We investigate through numerical simulations how a two-dimensional crystal yields and flows under an applied shear. We focus over a range that allows us to both address the response in the limit of an infinitesimal shear rate and describe the phase behavior of the system at a finite shear rate. In doing so, we carefully discuss the role of the topological defects and of the finite-size effects. We map out the whole phase diagram of the flowing steady state in the plane formed by temperature and shear rate. Shear-induced melting of the two-dimensional crystal is found to proceed in two steps: first, the solid loses long-range bond-orientational order and flows, even for an infinitesimal shear rate (in the thermodynamic limit). The resulting flowing hexatic phase then melts to a flowing, rather isotropic, liquid at a finite shear rate that depends on temperature. Finally, at a high shear rate, a third regime corresponding to a strongly anisotropic string-like flowing phase appears.