We consider minimal subshifts arising from substitutions on a finite alphabet. For example, iterating the substitution 0 --> 0111, 1 --> 0 one gets an infinite sequence 0111000011101110111... whose orbit closure forms such a subshift. Substitution systems are never strongly mixing (with respect to the unique invariant measure), but they may be topologically mixing, as shown by M. Dekking and M. Keane in 1978. If all the eigenvalues of the substitution matrix, except the largest one, are less than one in modulus, then the system has continuous eigenfunctions, hence it cannot be topologically mixing. In a joint work with R. Kenyon and L. Sadun, we prove that in the two-symbol case, if both eigenvalues are strictly greater than one in modulus (as in the example above), then the system is topologically mixing. This settles a conjecture of A. Livshits, who obtained partial results in this direction. If one of the eigenvalues equals 1 or -1, then the situation is more delicate: the system may or may not be topologically mixing.