In my household, peeling asparagus is considered one of those fancy chef's tricks and more often than not avoided. But many people differ and insist that asparagus should be peeled, the exception being the teen-tiny pencil thin spears because, well, then there would be nothing left.
The reason for peeling is that asparagus skins to be tough and stringy. The method is to place the asparagus flat on your cutting board and using a swivel peeler, shave thin sections of the peel off the stalk, stopping at the tips.
(Image: Dana Velden)