State-of-the-art global ocean circulation models used in climate studies are only passing the edge of becoming “eddy-permitting” or barely eddy-resolving. Such models commonly suffer from overdissipation of mesoscale eddies by routinely used subgrid dissipation (viscosity) operators and a resulting depletion of energy in the large-scale structures which are crucial for draining available potential energy into kinetic energy. More broadly, subgrid momentum closures may lead to both overdissipation or pileup of eddy kinetic energy and enstrophy of the smallest resolvable scales. The aim of this chapter is twofold. First, it reviews the theory of two-dimensional and geostrophic turbulence. To a large part, this is textbook material with particular emphasis, however, on issues relevant to modeling the global ocean in the eddy- permitting regime. Second, we discuss several recent parameterizations of subgrid dynamics, including simplified backscatter schemes by Jansen and Held, stochastic superparameterizations by Grooms and Majda, and an empirical backscatter scheme by Mana and Zanna.