Yes. I understand they're called garments. I was in Utah last summer and so many people were wearing knee length shorts. I was with a friend who was an ex-Mormon. He told me it was to cover their 'garments.' Only married Mormons wear them.

Only worthy mormons who have taken part in certain temple ceremonies. Not all mormons are welcome in the temple, but those who are take these things very seriously.

Moderator - the commenter is correct. Please delete this answer.

Yes, but only "Temple Mormons" which comprise approximately 20% of active Latter Day Saints. The slang term "Temple Mormon" means: "Those Mormons who been through an LDS Temple and have taken part in the Endowment ceremony." Thus "special underwear" is known as "Temple Garments". The article on WikiPedia on LDS Temple Garments is actually quite good. Here's a key excerpt: "[The Temple Garment] is worn as a religious vesture beneath the clothing of those who have taken part in the Endowment ceremony. The undergarments are viewed as a symbolic reminder of the sacred covenants made in temple ceremonies, and are viewed as an either symbolic or literal source of protection from the evils of the world." http://en.wikipedia.org/wiki/Temple_garments Please note, the endowment ceremony that Temple Garments play an important part in is another topic that is just too big to cover here! Unfortunately, understanding the "how's and why's" of Temple Garments requires an understanding of the Endowment Ceremony that they are a key component of. That end, I have provided a full set of links to sites that cover it in detail in my answer to this question: http://www.answerbag.com/a_view/5345099