Magic squares have fascinated people throughout centuries. The first references to magic squares can be found in ancient Chinese and Indian literature. The reader can find up-to-date information in many web-pages. A generalisation of a magic square is a magic p-dimensional cube. A magic p-dimensional cube of order n is a p-dimensional matrix containing natural numbers 1 ,..., n^p such that the sum of the numbers along every row and every diagonal is the same. Probably the first mentioned magic cube appeared in a P.Fermat's letter from 1640. Knowledge of magic p-dimensional cubes can find its use not only in amusing mathematics but also in many fields of mathematics and physics. In this paper we proved the first definitive result: Theorem. A magic p-dimensional cube of order n exists if and only if p≥ 2 and n≠2 or p = 1. The proof gives a simple algorithm for construction of a magic p-dimensional cube. A special case of the algorithm is a very simple method lo make magic squares. Mathematicians (and not only them) have studied many properties of magic squares and formulated problems which have not been solved. We can formulate similar problems for magic cubes, too.