We consider the Riemann–Hilbert (Hilbert) problem in classes similar to the Hardy class for general first-order elliptic systems on a plane. We establish basic properties ofHardy classes for solutions of that systems and solvability conditions for boundary-value problems. We construct the example demonstrating that for discontinuous coefficients the solvability features differ from the pictures of solvability of analogous problems for holomorphic and generalized analytic functions. In particular, the problem with positive index can be unsolvable.