Abstract: We will survey results on analytic regularity for degenerate elliptic operators. We recall classical results from the elliptic theory to motivate a discussion of the symplectic geometry of the characteristic set. Analytic hypoellipticity for "sum of squares" operators is the current focus of research. We discuss the Treves conjectures and give several examples. We present various notions of analytic hypoellipticity (AH) for operators with double characteristics. We will compare the following ideas: AH on open subsets of Euclidean space; global AH on manifolds; AH in the sense of germs.