  Two orthonormal bases B and B' of a d-dimensional complex inner-product space
are called mutually unbiased if and only if |<b|b'>|^2=1/d holds for all b in B
and b' in B'. The size of any set containing (pairwise) mutually unbiased bases
of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets
containing d+1 mutually unbiased bases are known to exist. We give a simplified
proof of this fact based on the estimation of exponential sums. We discuss
conjectures and open problems concerning the maximal number of mutually
unbiased bases for arbitrary dimensions.
