Let Bs be the set of those positive integers which can be written as a sum of s biquadrates. In 1939, H. Davenport showed that the complement of B16 in N is finite. The main result of the book states that every integer N > 10216, which is not divisible by 16, is contained in B16. It can be combined with results of a companion paper of Deshouillers, Hennecart and Landreau (which relies on heavy computations) to get the implication that every integer n > 13792 lies in B16 (this result is optimal, as 31.16m is not included in B15). The proof involves a combination of the Hardy-Littlewood method with several identities involving biquadrates and squares.