A statistical-mechanical treatment of equilibrium flows in the two-dimensional Euler fluid is constructed which respects all conservation laws. The vorticity field is fundamental, and its long-range Coulomb interactions lead to an exact set of nonlinear mean-field equations for the equilibrium state. The equations depend on an infinite set of parameters, in one-to-one correspondence with the infinite set of conserved variables. The equations contain all previous approximations as special limiting cases: for example the Kraichnan energy-enstrophy theory, and the Lundgren and Pointin point vortex mean-field theories are rederived. The techniques may be generalized to a number of other Coulomb-like Hamiltonian systems with an infinite number of conservation laws, including some in higher dimensions. For example, we rederive Lynden-Bell’s theory of stellar-cluster formation, as well as the Debye-Huckel theory of electrolytes. Our results may also be applicable to cylindrically bound guiding-center plasmas, which under idealized conditions provide another realization of 2-d Euler flow. Finally, a phenomenological theory of the weakly driven, weakly damped Euler fluid, based on weak perturbations of the equilibrium state, is presented. A very simple two-parameter model is used to illustrate the principal ideas.