In this paper, we introduce and analyze several new classes of multi-dimensional almost automorphic functions which generalize the classical one of Bochner. We develop the basic theory for the introduced classes, investigating the themes like composition principles, convolution invariance and the invariance under the actions of convolution products. We present several illustrative examples and applications to the abstract Volterra integro-differential equations and partial differential equations, providing also a mini appendix about almost automorphic functions on semi-topological groups.