Recent experiments on the anisotropic spin-1/2 triangular antiferromagnet Cs 2 CuBr 4 have revealed a remarkably rich phase diagram in applied magnetic fields, consisting of an unexpectedly large number of ordered phases. Motivated by this finding, we study the role of three ingredients-spatial anisotropy, Dzyaloshinskii-Moriya interactions, and quantum fluctuations-on the magnetization process of a triangular antiferromagnet, coming from the semiclassical limit. The richness of the problem stems from two key facts: (1) the classical isotropic model with a magnetic field exhibits a large accidental ground-state degeneracy and (2) these three ingredients compete with one another and split this degeneracy in opposing ways. Using a variety of complementary approaches, including extensive Monte Carlo numerics, spin-wave theory, and an analysis of Bose-Einstein condensation of magnons at high fields, we find that their interplay gives rise to a complex phase diagram consisting of numerous incommensurate and commensurate phases. Our results shed light on the observed phase diagram for Cs 2 CuBr 4 and suggest a number of future theoretical and experimental directions that will be useful for obtaining a complete understanding of this material's interesting phenomenology.