Mellon takes over from Michael Jackson, who failed to keep the club in League One after taking over from Graham Turner in January.
Jackson will remain on the coaching staff as assistant to former Fleetwood Town boss Mellon.
The 42-year-old takes charge on a one-year rolling contract and was quick to express excitement after agreeing the deal with the New Meadow outfit.
Mellon said: "It was important to me that I spoke to the chairman (Roland Wycherley) because I wanted to work for a guy like him who really loves the club and wants to move it forward. That matches my ambition in what I wanted to do.
"After looking at the environment we're going to be playing in and knowing I could keep lads like (assistant) Danny Coyne and Michael Jackson with me, which I'm delighted about because I know them really well, I'm looking forward to a really exciting adventure.
"There's a lot of work that needs to be done but we're not scared of hard work.
"We'll endeavour to put a team out on the pitch that represents the people of Shrewsbury, the Shrewsbury fans, and they come and enjoy watching it."
