  We present a new approach to the problem of mutually unbiased bases (MUBs),
based on positive definite functions on the unitary group. The method provides
a new proof of the fact that there are at most $d+1$ MUBs in ${\mathbb C}^d$.
It may also lead to a proof of non-existence of complete systems of MUBs in
dimension 6 via a conjectured algebraic identity.
