Explosions were heard overnight in the capital, Kyiv, where the mayor said five people had been injured in the "biggest" kamikaze drone attack so far.
One person was killed in the attack on the southern Odesa region. Ukraine's Red Cross says its warehouse was hit.
It marks the fourth attack in eight days on Kyiv and comes just 24 hours before Russia celebrates Victory Day.
The annual holiday commemorates the Soviet Union's victory over Nazi Germany during World War Two, a conflict the Kremlin has baselessly tried to draw parallels with since launching its invasion of Ukraine last year.
After a lull in Russian attacks on civilian targets in recent months, which saw Kyiv go days without an attack, Moscow has intensified its air raids over the past week ahead of a widely expected Ukrainian counter-offensive.
The Ukrainian military said the latest Russian raids - which lasted for more than four hours and were launched shortly after midnight - saw Iranian-made Shahed kamikaze drones swarm across the country.
Kyiv's Mayor Vitaliy Klitschko said nearly 60 drones had been launched by Russia, describing it as the "biggest" such attack so far.
He added that all 36 drones had been destroyed over Kyiv, but five people had been injured by falling debris from downed drones.
Emergency services responded after drone wreckage fell on a runway at Zhuliany international airport - one of the city's two commercial airports - Kyiv's military administration said.
And civilians were injured after drone debris hit a residential building in the central Shevchenkivskyi district, the administration added.
Elsewhere, in the Black Sea port city of Odesa, a warehouse was set ablaze after eight missiles were fired at targets by Russian bombers, Ukrainian officials said.
In a statement, Ukraine's Red Cross said its warehouse with humanitarian aid was destroyed and all aid deliveries had to be suspended.
Natalia Humeniuk, a spokesperson for Ukraine's Southern Command, later said a body of a man - a security guard - was pulled from the wreckage.
In a daily update, the Ukrainian military's command said there had also been a wave of missile strikes on the Kherson, Kharkiv and Mykolaiv regions.
At least eight people - including a child - were injured in two villages in the southern Kherson region, local officials said.
In Zaporizhzhia, the head of the Russian installed administration, Vladimir Rogov, said Russian forces hit a warehouse and a Ukrainian troop position in the small city of Orikhiv.
On the eastern front, the Ukrainian commander of forces in the besieged eastern city of Bakhmut said Russian troops had stepped up shelling, in a bid to take the city by Tuesday's celebrations.
Russian troops and fighters from the Wagner Group, a private military company, have been trying to capture Bakhmut for months - despite its questionable strategic value.
Over the weekend, Wagner's founder Yevgeny Prigozhin appeared to U-turn on a threat to withdraw from the city after he was promised fresh ammunition supplies by the defence ministry in Moscow.
Meanwhile, Ukrainian President Volodymyr Zelensky announced that 9 May would from now be celebrated as Europe Day, in line with the European Union. The move - which needs parliamentary approval - is seen as a pointed rebuke to Russia.
Mr Zelensky said he had signed a decree that the day would commemorate European unity and the defeat of "Ruscism" - a term that is shorthand for "Russian fascism".
He also said that 8 May would now officially be a Day of Remembrance and Victory, as marked in many countries around the world.
bbc