Abstract: We present a systematic development of energy-stable approximations of the two-dimensional shallow water (SW) equations, which are based on the general framework of entropy conservative schemes introduced in [E. Tadmor, Acta Numerica 12, (2003) pp. 451-512] and [E. Tadmor and W. Zhong, JHDEs 3 (2006) pp. 529-559]. No artificial numerical viscosity is involved: stability is dictated solely by eddy viscosity. In particular, in the absence of any dissipative mechanism, the resulting numerical schemes precisely preserve the total energy, which serves as an entropy function for the SW equations. We demonstrate the dispersive nature of such entropy conservative schemes with a series of scalar examples, interesting for their own sake. We then turn to the SW equations. Numerical experiments of the partial-dam-break problem with energy-preserving and energy stable schemes, successfully simulate the propagation of circular shock and the vortices formed on the both sides of the breach.