We study mirror symmetry for Miles Reid's 95 families of Q-Fano 3-folds Y. In accordance with mirror symmetry predictions, we show that the mirror of the log Calabi-Yau pair (Y,E) where E is an aticanonical divisor in |-KY| is a pair (X,D) together with a morphism W to the projective line whose general fiber F is a K3 surface and has exactly 3 singular fibers. We describe the general fiber of W explicitly, for instance, we give a formula for the Picard rank of F in terms of the singularities of Y. Finally, we use the theory of hypergeometric groups to describe the monodromy of the mirror family. In particular, we show that a power of the monodromy at infinitiy is maximally unipotent. This allows us to conclude that the fiber D over infinity after a base change and birational modifications yields a K3 surface of type III in Kulikov's notation.