A particle moves in circular arcs with Larmor radius R between specular reflections at the smooth convex boundary of a planar region. The dynamics depends on the value of R in relation to the extreme curvature radii rho _min and rho _max and the radius R* of the largest circle that can be inscribed in the boundary. For R<R* some orbits are complete Larmor circles and constitute an integrable component of the motion; all other orbits bounce repeatedly. For rho _min<R< rho _max there are 'flyaway intervals' on the boundary for which glancing orbits are a powerful source of chaos in the map (on the phase cylinder) relating successive bounces; this type of chaos is a characteristic feature of magnetic billiards. For sufficiently large R the simplest closed orbits consist of two arcs associated with diameters of the boundary; their existence and stability can be determined. In several regimes where motion consists of short skips between nearby boundary points (including the strong-field case R to 0), an explicit adiabatic invariant can be found which gives an excellent approximation to the exact invariant curves in these regimes. Computations for a magnetic billiard with elliptic boundary illustrate the theory.