Models of solar flares and coronal mass ejections typically predict the development of an elongated current sheet in the wake behind the rising flux rope. In reality, reconnection in these current sheets will be asymmetric along the inflow, outflow, and out-of-plane directions. We perform resistive MHD simulations to investigate the consequences of asymmetry during solar reconnection. We predict several observational signatures of asymmetric reconnection, including flare loops with a skewed candle flame shape, slow drifting of the current sheet into the strong field upstream region, asymmetric footpoint speeds and hard X-ray emission, and rolling motions within the erupting flux rope. There is net plasma flow across the magnetic field null along both the inflow and outflow directions. We compare simulations to SDO/AIA, Hinode/XRT, and STEREO observations of flare loop shapes, current sheet drifting, and rolling motions during prominence eruptions. Simulations of the plasmoid instability with different upstream magnetic fields show that the reconnection rate remains enhanced even during the asymmetric case. The islands preferentially grow into the weak field upstream region. The islands develop net vorticity because the outflow jets impact them obliquely rather than directly. Asymmetric reconnection in the chromosphere occurs when emerging flux interacts with pre-existing overlying flux. We present initial results on asymmetric reconnection in partially ionized chromospheric plasmas. Finally, we discuss how comparisons to observations are necessary to understand the role of three-dimensional effects.