By 1939 it was known that 13792 cannot be expressed as a sum of sixteen biquadrates (folklore), that there exist infinitely many natural numbers which cannot be written as sums of fifteen biquadrates (Kempner) and that every sufficiently large integer is a sum of sixteen biquadrates (Davenport). In this memoir it is shown that every integer larger than $10^{216}$ and not divisible by 16 can be represented as a sum of sixteen biquadrates. Combined with a numerical study by Deshouillers, Hennecart and Landreau, this result implies that every integer larger than 13792 is a sum of sixteen biquadrates.