We have applied the minimum variance (MV) adaptive beamformer to medical ultrasound imaging and shown significant improvement in image quality compared to delay-and-sum (DAS). We demonstrate reduced mainlobe width and suppression of sidelobes on both simulated and experimental RF data of closely spaced wire targets, which gives potential contrast and resolution enhancement in medical images. The method is applied to experimental RF data from a heart phantom, in which we show increased resolution and improved definition of the ventricular walls. A potential weakness of adaptive beamformers is sensitivity to errors in the assumed wavefield parameters. We look at two ways to increase robustness of the proposed method; spatial smoothing and diagonal loading. We show that both are controlled by a single parameter that can move the performance from that of a MV beamformer to that of a DAS beamformer. We evaluate the sensitivity to velocity errors and show that reliable amplitude estimates are achieved while the mainlobe width and sidelobe levels are still significantly lower than for the conventional beamformer.