We study the well-posedness and convergence of column-and-constraint generation algorithms for general two-stage robust optimization problems. The analysis is formulated in terms of regularity properties of the objective, the second-stage feasible region mapping, and the separation value function, without relying on a particular algebraic representation of the second-stage problem. We give sufficient conditions for the existence of solutions and the well-posedness of the algorithm, prove finite termination for every fixed positive tolerance, and characterize accumulation points when zero tolerance is used. We then introduce pointwise, graph, and violation tolerance properties, which translate the stopping condition into guarantees on the objective and approximate second-stage feasibility. The resulting theory is applied to canonical, residual-based, and graph-distance separation value functions. Finally, we consider robust bilevel optimization with a wait-and-see convex follower and obtain finite-termination and convergence guarantees for the corresponding algorithm under explicit regularity assumptions.