In the present work, we introduce the concept of asymptotically almost automorphic functions on time scales and study their main properties. We study nonautonomous dynamic equations on time scales given by x(Delta)(t) = A(t)x(t) + f(t) and x(Delta)(t) = A(t)x(t) + f(t, x(t)), t is an element of T where T is an invariant under translations time scale and A is an element of R(T, R-nxn). We give new criteria ensuring the existence of an asymptotically almost automorphic solution for both equations.