Taylor, who had a year remaining on his contract with the Shrews, has put pen to paper on a three-year deal.
The 21-year-old former Wigan trainee was coveted by several League One clubs but Posh have won the race for his signature despite last-minute advances from their rivals, according to chairman Darragh MacAnthony.
"Delighted to announce the transfer of young J Taylor .Can play wide right, left and top of a diamond and will add pace, aggressiveness, goals and assists to our midfield next season," MacAnthony said via his Twitter account.
"Also commiserations other 2 top-6 L1 clubs who tried to hijack the deal this morning. Thanks to Shrewsbury, who have been brilliant to deal with in making this happen.
"Past successful Posh sides have been built on flair, goals and assists from our attacking midfielders and this was an area we lacked last season.More to come!"
Taylor joined Shrewsbury in 2008 after leaving Wigan. He scored nine league goals last season for a Town team that suffered relegation from League One, including a strike against Peterborough in Posh's 4-2 win at Greanhous Meadow in April.