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In this article we study percolation on the Cayley graph of a free product ofgroups.  The critical probability $p_c$ of a free product $G_1*G_2*...*G_n$ of groupsis found as a solution of an equation involving only the expected subcriticalcluster size of factor groups $G_1,G_2,...,G_n$. For finite groups theseequations are polynomial and can be explicitly written down. The expectedsubcritical cluster size of the free product is also found in terms of thesubcritical cluster sizes of the factors. In particular, we prove that $p_c$for the Cayley graph of the modular group $\hbox{PSL}_2(\mathbb Z)$ (with thestandard generators) is $.5199...$, the unique root of the polynomial$2p^5-6p^4+2p^3+4p^2-1$ in the interval $(0,1)$.  In the case when groups $G_i$ can be "well approximated" by a sequence ofquotient groups, we show that the critical probabilities of the free product ofthese approximations converge to the critical probability of $G_1*G_2*...*G_n$and the speed of convergence is exponential. Thus for residually finite groups,for example, one can restrict oneself to the case when each free factor isfinite.  We show that the critical point, introduced by Schonmann, $p_{\mathrm{exp}}$of the free product is just the minimum of $p_{\mathrm{exp}}$ for the factors

CRITICAL PERCOLATION OF FREE PRODUCT OF GROUPS