Godden was Boro’s leading goalscorer last season, scoring 14 times in 44 appearances in all competitions.
The 26-year-old had one year left to run on his contract with Stevenage, but rejected two offers of a new deal with the League Two club.
— Peterborough United (@theposhofficial) June 4, 2018
“I’m delighted – I just want to push on now, help the club go places and score some goals,” Godden said.
He is the sixth signing since the end of last season for Peterborough, who finished ninth in League One.