In Kyiv, smoke and flames were seen coming from a tower block, as fire crews worked to tackle the fire.
“I was scared when air alerts were announced and we rushed here,” Tetyana, 49, said as she hugged her two-year-old granddaughter and a small dog in a bomb shelter in central Kyiv.
She added: “I hope they will shoot down all of them. I pray for our air defence.”
Ukrainian president Volodymyr Zelenskyy said six regions came under enemy attack.
“In Kyiv, more than ten people were injured, as of now, two deaths are known,” he said on Telegram.
“There may be people under the rubble.”
President Zelenskyy also vowed that “terrorists will always feel the consequences of their actions” and paid tribute to the medical and rescue workers.
Parts of Kyiv were without power following the attack, after a downed Russian missile damaged power lines, Mayor Vitali Klitschko said.
The comments from President Zelenskyy and Mr Zaluzhnyi come amid reports of a rift between them.