The 20-year-old winger has signed a three-year-contract with the Railwaymen after the player caught the management teams eye with his perfomances for Aldershot last season.
Grant made over 40 appearanaces for the Shots and also made appearances for the U-23 Jamacia team last term.
Grant spoke to the official Crewe website 'I like the way that Crewe works and how they bring on players and it all feels really homely here. I'm delighted about the move and I hope to do my best here.
'I've played here before so I know all about the supporters. I don't know the ins and outs of the club but I do know that it's a good gamble for me.
'I know a few people who used to play here, a few of my team-mates at Aldershot used to play at Crewe, so hopefully it will be the same.'