Note: Shrimp need to be kept cold. While you are working with them, keep them in on ice, or in ice water. See this link about seafood safety.
1 Pull off the head (if it is still attached) and legs.
2 Starting with the head end, pull off the outer shell. Depending on how you intend to present the shrimp, you can keep the last segment of shell and the tail tip on, for decorative purposes. Place shells in a plastic bag, securely close, and either discard or freeze for making shellfish stock.
Alternatively, you can leave the shell on and use a pair of kitchen scissors to cut along the outer edge of the shrimp's back, cutting the shell so you can get to the vein. The shells hold a lot of flavor, so there is something to be said for cooking shrimp with their shells on.
3 Using a small paring knife, cut along the outer edge of the shrimp's back, about 1/4-inch deep.
4 If you can see it, remove and discard the vein the runs along right under the surface of the back, with your fingers or the tip of your knife. If you can't see the vein, don't bother with it.
Return the peeled and deveined shrimp to your bowl of ice or ice water until you are ready to cook.