We introduce and study several affine (=annular in this paper) versions of the classical diagram algebras such as Temperley-Lieb, partition, Brauer, Motzkin, rook Brauer, rook, planar partition, and planar rook algebras. We give generators and relation presentation for them and their associated categories, study their representation theory, and the asymptotic behavior of tensor products of their representations in the monoid case. Under a mild hypothesis, we also prove a previous conjecture concerning the asymptotic growth of the number of indecomposable summands in the tensor powers of representations for finite monoids.