The talk will focus on joint distributions of non-commuting random variables. After introducing noncommutative distributions and showing how they can be realized as distributions of concrete random variables, we will show how they can be studied with the aid of free noncommutative analytic transforms. As an illustration of the power of this tool, we will provide an explicit characterization of the atomic part of the distribution of a selfadjoint polynomial evaluated in a tuple of free random variables (joint work with H. Bercovici and W. Liu), as well as some extensions of this result coming out of ongoing joint work with V. Vinnikov.