We give the first polylogarithmic-competitive randomized online algorithm for the
k
-server problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of Õ(log
3
n
log
2
k
) for any metric space on
n
points. Our algorithm improves upon the deterministic (2
k
-1)-competitive algorithm of Koutsoupias and Papadimitriou [Koutsoupias and Papadimitriou 1995] for a wide range of
n
.