Laser-induced cavitation bubbles near a curved rigid boundary are observed experimentally using high-speed photography. An image theory is applied to obtain information on global bubble motion while a boundary integral method is employed to gain a more detailed understanding of the behaviour of a liquid jet that threads a collapsing bubble, creating a toroidal bubble. Comparisons between the theory and experiment show that when a comparable sized bubble is located near a rigid boundary the bubble motion is significantly influenced by the surface curvature of the boundary, which is characterized by a parameter ζ, giving convex walls for ζ < 1, concave walls for ζ > 1 and a flat wall when ζ = 1. If a boundary is slightly concave, the most pronounced migration occurs at the first bubble collapse. The velocity of a liquid jet impacting on the far side of the bubble surface tends to increase with decreasing parameter ζ. In the case of a convex boundary, the jet velocity is larger than that generated in the flat boundary case. Although the situation considered here is restricted to axisymmetric motion without mean flow, this result suggests that higher pressures can occur when cavitation bubbles collapse near a non-flat boundary. Bubble separation, including the pinch-off phenomenon, is observed in the final stage of the collapse of a bubble, with the oblate shape at its maximum volume attached to the surface of a convex boundary, followed by bubble splitting which is responsible for further bubble proliferation.