We examine the low-energy behavior of two-dimensional electrons with current-current interactions. The interactions are long range, cannot be screened, and result in anomalies in perturbation theory. In the Hartree-Fock theory, we find an instability of the Landau Fermi-liquid state resulting in ``fermion-condensate'' states with a high degeneracy of levels at the Fermi energy. This degeneracy may be removed to give a Fermi-liquid state with vanishing Fermi velocity. As an example, we investigate further the Fermi-liquid state with the quasiparticle spectrum ɛ(p)~(p-pF)3/2 proposed by Halperin et al. Application of the recent Haldane scheme of bosonization in higher dimensions to this state leads to coupling between bosons, which has the form required for a breakdown of the Landau Fermi-liquid behavior. A generalized form of the Luttinger liquid is obtained.