In this paper, we present and analyze several new types of almost periodic functions in several variables, namely: (RX,B)-almost periodic functions, Bohr B-almost periodic functions, uniformly recurrent B-almost periodic functions, strongly B-almost periodic functions, where RX is a non-empty collection of sequences in Rn×X, B denotes a non-empty collection of non-empty subsets of X, and X is a Banach space. We present their principal structural characterizations, their composition principles and invariance under convolution products. After that, we provide several applications of our abstract theoretical results to partial differential equations and to the abstract integral equations in Banach spaces.