  The two-dimensional shallow water equations in Eulerian and Lagrangain
coordinates are considered. Lagrangian and Hamiltonian formalism of the
equations is given. The transformations mapping the two-dimensional shallow
water equations with a circular or plane bottom into the gas dynamics equations
of a polytropic gas with polytropic exponent $\gamma=2$ is represented.
  Group properties of the equations are considered, and the group
classification for the case of the elliptic paraboloid bottom topography is
performed.
  The properties of the two-dimensional shallow water equations in Lagrangian
coordinates are discussed from the discretization point of view. New invariant
conservative finite-difference schemes for the equations and their
one-dimensional reductions are constructed. The schemes are derived either by
extending the known one-dimensional schemes or by direct algebraic construction
based on some assumptions on the form of the energy conservation law. Among the
proposed schemes there are schemes possessing conservation laws of mass and
energy.
