Based on Jay Ward's cartoon characters from Peabody's Improbable History on the 60s television show Rocky and Bullwinkle, Mr Peabody is the world's smartest person ... who just happens to be a dog.
Rejected by human owners, he turns his mind to amazing accomplishments, including the invention of the WABAC (way back) time machine. He also finds an unwanted baby, Sherman, and adopts him.
On Sherman's first day of school he gets into a fight, bullied by schoolmate Penny, who calls him a dog.
When child services threaten to take Sherman away, Mr Peabody attempts to win them around, but it all turns hilariously chaotic when Sherman takes Penny in the time machine.
Travelling through the Renaissance and the Battle of Troy, and inspiring the painting of the Mona Lisa, they need to enlist Mr Peabody's help. Hiccups in their time travel lead to history spinning out of control, and Mr Peabody and Sherman must work out how to save the world.