An algebra over a field is called affine if it is finitely generated and is a central order in some finite dimensional central simple algebra. In this paper the elementary properties of such algebras are studied first. Then the concept of affine Krull dimension is introduced, and an analogue of the well-known Markov-Procesi theorem on the equality of the affine Krull dimension and the degree of transcendence of the centroid of an affine algebra is proved. Next free affine algebras are defined, and then the Krull dimensions are computed for those connected with a Cayley- Dickson algebra, the simple non-Lie-Malcev algebra, and the Jordan algebra of a bilinear form.