In this chapter, two types of the multi-dimensional classical Hardy spaces, namely the Hp □(ℝd) and Hp(ℝd) spaces, are introduced. All the results of Chap. 1, amongst others, inequalities, atomic decompositions, interpolation theorems, boundedness results are proved for these spaces. Basically, the results for Hp □(ℝd) are very similar to those for the one-dimensional Hp(ℝ) spaces studied in Chap. 1, so we omit the corresponding proofs. However, the proofs for Hp(ℝd) are different from the one-dimensional version requiring new ideas. We also study some generalizations of the Hardy-Littlewood maximal function for multi-dimensional functions.