  We introduce Macdonald polynomials indexed by $n$-tuples of partitions and
characterized by certain orthogonality and triangularity relations. We prove
that they can be explicitly given as products of ordinary Macdonald polynomials
depending on special alphabets. With this factorization in hand, we establish
their most basic properties, such as explicit formulas for their norm-squared,
evaluation and reproducing kernel. Moreover, we show that the $q,t$-Kostka
coefficients associated to the multi-Macdonald polynomials are positive and
correspond to $q,t$-analogs of the dimensions of the irreducible
representations of $C_n \sim S_d$, the wreath product of the cyclic group $C_n$
with the symmetric group.
