We derive subordination functions for free additive and free multiplicative deconvolutions under mild moment conditions. Our results include an algorithm to calculate these subordination functions, and thus the associated Cauchy transforms, for complex numbers with imaginary part greater than a parameter depending on the measure to deconvolve. The existence of these subordination functions on such domains reduces the problem of free deconvolutions to the problem of the classical additive deconvolution with a Cauchy distribution. Thus, our results, combined with known methods for the deconvolution with a Cauchy distribution, allow us to solve the free deconvolution problem. We also present extensions of these results to the case of operator-valued deconvolutions.