We show that the canonical random-cluster measure associated to isoradial graphs is critical for all q \geq 1. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for 1 \leq q \leq 4 and discontinuous for q>4. For 1 \leq q \leq 4, the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular, these properties also hold on the triangular and hexagonal lattices. Our results also include the limiting case of quantum random-cluster models in 1+1 dimensions.