  The variety of sums of powers of a homogeneous polynomial of degree d in n
variables is defined and investigated in some examples, old and new. These
varieties are studied via apolarity and syzygies. Classical results of
Sylvester (1851), Hilbert (1888), Dixon and Stuart (1906) and some more recent
results of Mukai (1992) are presented together with new results for the cases
(n,d)=(3,8), (4,2), (5,3). In the last case the variety of sums of 8 powers of
a general cubic form is a Fano 5-fold of index 1 and degree 660.

Algebraic Geometry