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In this paper, a novel linearly constrained robust Capon beamformer (LCRCB) framework is proposed. In the LCRCB, linear constraints can be used, e.g., for beampattern control and ellipsoidal array steering vector sets can be exploited, using robust Capon beamforming techniques, e.g., to allow for arbitrary array steering vector errors, such as those arising from calibration errors. The LCRCB is applicable to arbitrary array geometries and can be computed efficiently. For the limiting case that the ellipsoid is a point, we show that the LCRCB coincides with a linearly constrained minimum variance beamformer. To show the utility of the LCRCB, mainbeam and null-pattern control examples are included.