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Subject of this thesis is the issue of equal-order finite element discretization of hydrostatic flow problems. These flow problems typically arise in geophysical fluid dynamics on large scales and in flat domains. This small aspect ratio between the depth and the horizontal extents of the considered domain allows to efficiently reduce the complexity of the incompressible three dimensional Navier-Stokes equations, which form the basis of geophysical flows. In the resulting set of equations, the vertical momentum equation is replaced by the hydrostatic balance, which thus decouples the vertical pressure variations from the dynamic system, and the dynamically relevant pressure becomes two dimensional. Moreover, the vertical velocity component can be explicitely determined by the horizontal velocity components. Concomitant with this reduction is the replacement of the divergence constraint by a suitably modified version of it. As in the classical framework, it is known that these hydrostatic flow problems also show a saddle point structure, and there is a similar uncertainty concerning existence and uniqueness of solutions as is apparent for the classical case. Although the variational framework has been intensively treated, the issue of the discretization, in particular the finite element discretization of hydrostatic problems has hardly been considered yet. The present work dedicates to this topic. We indicate the tight relation between a finite element discretized hydrostatic flow problem and its two dimensional counterpart with respect to inf-sup stability. Moreover, we elaborate stabilization techniques in order to result to inf-sup stable schemes and to suitably treat the case of dominant advection. For each of these cases we can draw on classical stabilization schemes. For the isotropic hydrostatic Stokes problem we thus derive and examine residual-based as well as symmetric stabilization schemes. In the appropriate Oseen case we restrict to symmetric stabilization schemes. Beside the isotropic case, we also ...