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The Adamastache is the facial hair adorning Admiral William Adama's upper lip.
The Adamastache first appears to us in "Scattered", where it was revealed that Adama wore it during his civilian career. It appears again after the one-year jump in "Lay Down Your Burdens, Part II" but meets an untimely demise at the beginning of Season Three in the episode "Exodus, Part II". While the shaving of the Adamastache was possibly symbolic of a new beginning for the Colonies, we also learn through flashbacks in the episode "Unfinished Business" that Adama wasted no time in reacquiring the Adamastache, growing it within a month of arrival at New Caprica. As such, it was party to a "laid-back" Adama and possibly played a role in the alleged nookie between the Admiral and Laura Roslin on New Caprica.
A photograph of Starbuck with an Adamastache can be seen in "The Son Also Rises". Given the Adamastache's apparent ability to be mysteriously resurrected after being killed by shaving, and the obvious presence of more than one copy of it, the moustache may very well be the final Cylon model.
A moustache has historically indicated the man wearing it was a soldier. As Galactica and Pegasus had jumped away from the civilians, Adama's role during the time immediately before he shaved it was purely military. Though clearly he didn't originally grow it to symbolize that he was solely a military man without the civilian population, shaving it could still serve as a sign that, with civil authority restored, he is again bound by more than military objectives. Also, it helped get the nookie when he didn't see Roslin so often, but, now that they interact professionally and frequently once more, it is no longer needed.
Not to be confused with Adama's Stash.