Let X be an irreducible and reduced projective variety in projective n-space over an algebraically closed field, and assume that X is not contained in a hyperplane. Then deg X > codim X. It is known that the varieties X of minimal degree (deg X = codim X + 1) have a linear minimal free resolution. Here we describe the structure of the minimal free resolution of a variety X of almost minimal degree (deg X = codim X + 2) of codim X < 5 by listing the possible Betti diagrams. This is joint work with M. Brodmann.