This has been CNBC's live blog covering updates on the war in Ukraine.

NATO allies are left to deliberate Ukraine's membership to the alliance, following a major European security summit in Moldova that saw Ukrainian President Volodymyr Zelenskyy push for his country's ability to join the Western defense organization.

The prospect of Ukraine's NATO membership is a contentious one over which member states are sharply divided, with many fearing further retaliation from Moscow. It is virtually impossible for a state to join NATO while that country is currently at war or has part of its territory occupied by a non-NATO adversarial state, given the risk that would pose to the rest of the members.

Meanwhile, Kyiv's military reports it shot down more than 30 Russian missiles and drones in an early morning attack, and Russian authorities in the country's western border regions say some of their villages have been shelled by Ukrainian forces.

White House confident Ukraine has stocked arsenals for potential counteroffensive

The White House said that it was confident that Ukraine is ready for its counteroffensive and has an adequate supply of air defense systems, armor and ammunition.

"We've done a lot of work to make sure we've filled out their shopping lists," National security council spokesman John Kirby told reporters on a conference call.

"They are going to expend some of this stuff in combat," Kirby said, adding that the U.S. and Western allies are "poised and ready through the remainder of this year" to provide Ukraine with additional weapons systems.

Kirby declined to speculate on when a Ukrainian counteroffensive will begin.

— Amanda Macias

One ship leaves Ukraine under Black Sea Grain Initiative

One ship left Ukraine's port of Odesa carrying 67,500 metric tons of corn under the Black Sea Grain Initiative, a deal that guarantees the export of agricultural goods amid Russia's ongoing war.

The Marshall Islands-flagged vessel, dubbed Zermatt, is destined for China, according to the organization responsible for tracking the export movements.

The Black Sea grain deal was extended last month but is set to expire in mid-July unless Russia, Ukraine, Turkey and the United Nations decide to draw out the agreement.

The first vessel under the deal left Ukraine's port of Odesa on Aug. 1, carrying more than 26,000 metric tons of corn.

— Amanda Macias

Ukraine military chief shares details of Russia's morning attack on Kyiv

The Ukrainian armed forces' commander-in-chief posted an update with more details on the early morning missile attack on Kyiv, describing the weapons used in a Telegram post.

"The occupiers continue to terrorize Ukraine with attack drones and missiles," Valerii Zaluzhnyi wrote.

"Around 11pm the enemy attacked Kyiv with Shahed-136/131 kamikaze drones. They entered from the southern direction, using the topography of the area and the course of the Dnieper River."

"At about 3am, Kh-101/Kh-555 cruise missiles launched by Tu-95ms strategic bombers from the Caspian Sea were already in the airspace of Ukraine," he wrote. "They entered from the northern direction, attacking Kyiv, maneuvering, trying to mislead our air defense."

He added that "All 15 cruise missiles and 21 attack drones were destroyed by the forces and means of air defense of the air force, in cooperation with the air defense systems of the defense forces of Ukraine."

CNBC was not able to independently verify the information. Ukraine's government regularly says it shoots down all or most of Russia's missiles and drones.

— Natasha Turak

Ukrainian city of Nikopol shelled overnight, regional governor says

The city of Nikopol in the southern Dnipropetrovsk region of Ukraine came under heavy shelling by Russian forces overnight, regional governor Serhiy Lysak reported.

"The aggressor does not stop. Nikopol region came under attack again. At midnight, the Rashists shelled Nikopol," Lysak wrote on his Telegram channel, using a term many Ukrainians use to refer to Russian armed forces members.

"Shells from heavy artillery flew into the city," he wrote. "People are unharmed. Rescuers are examining the area. The enemy is insidious and does not abandon its tactics of terrorizing the civilian population."

— Natasha Turak

U.S. will stop sharing certain New Start arms control data with Russia

The U.S. will stop sharing some data that is required to be shared with Russia under the New Start arms control treaty, the State Department said in a statement, in response to what it described as Russia's "ongoing violations" of the agreement.

The data that the U.S. will now withhold includes information on missile and launcher locations, and launches of American submarine-launched and intercontinental ballistic missiles.

"The United States is committed to full and mutual implementation of the New START Treaty. Consistent with that commitment, the United States has adopted lawful countermeasures in response to the Russian Federation's ongoing violations of the New START Treaty," the State Department statement said. "The Russian Federation's purported suspension of the New START Treaty is legally invalid."

It added that Washington's countermeasures "are fully consistent with international law" and are "proportionate, reversible, and meet all other legal requirements," and that it notified Russia in advance. The U.S. "remains ready to work constructively with Russia on resuming implementation" of the treaty, it said.  

The New Start Treaty, which puts limits on deployed nuclear arsenals, is essentially the last standing arms control agreement between the U.S. and Russia, after others have been dismantled or abandoned by one or both countries in the last several years. Moscow has not formally left the treaty, but in February decided to suspend its participation in it.

— Natasha Turak

Kyiv air defenses shot down more than 30 Russian missiles and drones, military says

Kyiv's air defenses shot down more than 30 drone and missiles strikes during an early morning attack by Russia, Ukrainian authorities said.

"According to preliminary information, more than 30 air targets of various types were detected and destroyed in the airspace over and around Kyiv by air defense forces," Kyiv's military authorities wrote in a post on Telegram. CNBC could not independently verify the information.

The city's mayor Vitali Klitschko wrote on Telegram that no call for rescue services had been made during the attacks.

— Natasha Turak

U.S. debt ceiling deal won't threaten spending for Ukraine, Sens. Schumer and McConnell say

U.S. lawmakers reached a last-minute deal to to raise the country's debt ceiling and cap government spending for two years, but that won't threaten Washington's financial and military support for Ukraine and other allies, Senate Majority Leader Chuck Schumer and Minority Leader Mitch McConnell said.

"This debt ceiling deal does nothing to limit the Senate's ability to appropriate emergency supplemental funds to ensure our military capabilities are sufficient to deter China, Russia and our other adversaries," the joint statement said, in response to concerns by more hawkish Republicans in Congress that the budget deal doesn't enable sufficient funding for the Pentagon.

"The Senate is not about to ignore our national needs, nor abandon our friends and allies who face urgent threats from America's most dangerous adversaries," the statement added.

President Joe Biden is expected to sign the legislation Friday, preventing what would have been the first ever U.S. debt default, which would have wreaked havoc on global financial markets.

— Natasha Turak

China's envoy to Russia says Moscow appreciates its efforts toward peace

Russia appreciates China's "sincere desire and earnest efforts" to find a solution to the war in Ukraine, China's special envoy for Eurasian affairs said during a press briefing.

"The risk of escalation of the Russia-Ukraine war is still high ... All sides must ensure the safety of nuclear facilities and take concrete measures to cool down the temperature," envoy Li Hui, said, according to Reuters. During the briefing, he discussed his diplomatic 12-day tour of Kyiv, Moscow, Warsaw, Paris, Brussels and Berlin aimed at seeking solutions to the conflict.  

Li said that he believes neither Ukraine nor Russia has "firmly" shut the door on peace negotiations, and added that, "as long as it's conducive to easing the situation, China is willing to do anything."

China has previously presented a multi-point plan for peace between Ukraine and Russia, which has drawn much criticism from Kyiv and its Western allies for being too accommodating to Russia. While Beijing has called for an end to the war, it has refrained from condemning or blaming Russia and much of its state-run press is highly favorable toward Moscow.

— Natasha Turak

Villages in Russia's western border region shelled, governor says

Villages in Russia's western Bryansk region were shelled, regional governor Alexander Bogomaz wrote on his Telegram channel, saying that the attackers were Ukrainian. He identified the two hit villages as Lomakovka and Novaya Pogoshch, saying that one house was set on fire, but that there were no deaths, according to Reuters reporting.

CNBC could not independently verify the claims.

Reported offensives on Russian soil have jumped in recent weeks, with drone attacks taking place in Moscow and shelling and incursions into Russia's western region that borders Ukraine. The Kremlin blames Ukraine's government for directing the hostilities, while Kyiv denies involvement.

Most of the recent attacks and attempted incursions into Russian territory have been claimed by the Freedom of Russia Legion and Russian Volunteer Corps, which describe themselves as pro-Ukraine ethnic Russians fighting to combat Russia's government.

— Natasha Turak

Zelenskyy presses hard for NATO membership at Moldova conference

Ukrainian President Volodymyr Zelenskyy pressed NATO to allow Ukraine's membership of the alliance — a topic of contention that has divided the group of 31 countries.

"This year is for decisions," Zelenskyy said at the European Political Community summit in Moldova on Thursday. "In summer in Vilnius at the NATO summit, a clear invitation from members of Ukraine is needed, and security guarantees on the way to NATO membership are needed."

Ukraine and those who support its bid say that NATO membership is needed to deter Russia from attacking the country again, while others warn that the move could further provoke Moscow, which sees Kyiv joining the Western alliance as a major red line.

Zelenskyy also made a push for EU membership, with EU Commission President Ursula von der Leyen expressing strong support. Ukraine would still need to meet a set of standards, including weeding out its well-documented corruption, in order to join the economic bloc.

— Natasha Turak

Russian forces target food lines and water infrastructure to starve Ukrainians, international lawyers say in a new report

Russian occupying forces in Ukraine employ starvation tactics on civilians by targeting food lines, agricultural harvests and water infrastructure, according to a team of international lawyers helping Kyiv investigate alleged war crimes. 

The investigators focused their efforts on Chernihiv, which was under siege for a little over two months before Russian troops were expelled from the northern Ukrainian city.

Catriona Murdoch, a lawyer and expert in starvation-related crimes, described Chernihiv as the "tip of the iceberg in Putin's calculated plan to terrorize, subjugate and kill Ukrainian people."

The Kremlin has previously denied that its forces commit war crimes or deliberately target civilians and related critical infrastructure.

Read the full story here.

— Amanda Macias

More than 500 Ukrainian children have died since start of Russia's war, U.S. says

U.S. Ambassador to Ukraine Bridget Brink wrote on Twitter that more than 500 Ukrainian children have died as a result of Russia's ongoing war.

"Heartbreaking: More than 500 Ukrainian children have been killed as a result of Russia's unprovoked, full-scale war in Ukraine," adding that "Two more children were tragically killed during another round of Russia's cowardly nighttime attacks."  

The Kremlin's widespread and systematic attacks against Ukraine's civilian population & its unlawful transfer and deportation of thousands of Ukrainian children are horrific examples of Russia's war crimes and crimes against humanity. Justice must be served.

— Ambassador Bridget A. Brink (@USAmbKyiv) June 1, 2023

"The Kremlin's widespread and systematic attacks against Ukraine's civilian population & its unlawful transfer and deportation of thousands of Ukrainian children are horrific examples of Russia's war crimes and crimes against humanity. Justice must be served," Brink added in a second tweet.

The Kremlin has previously denied that its troops in Ukraine engage in war crimes.

In March, the International Criminal Court issued an arrest warrant for Russian President Vladimir Putin and Maria Lvova-Belova, Russia's commissioner for children's rights.

Putin and Lvova-Belova are "allegedly responsible for the war crime of unlawful deportation" of children from Ukraine to Russia, the court wrote in a statement.

— Amanda Macias

Read CNBC's previous live coverage here:

Russia says it repelled another attack on its soil; UK PM says Ukraine's rightful place is in NATO