We present the zero temperature phase diagram of a Heisenberg antiferromagnet on a frustrated triangular lattice with nearest neighbor (J 1 ) and next nearest neighbor (J 2 ) interactions, in a magnetic field. We show that the classical model has an accidental degeneracy for all J 2 /J 1 and all fields, but the degeneracy is lifted by quantum fluctuations. We show that at large S, for J 2 /J 1 < 1/8, quantum fluctuations select the same sequence of three sublattice co-planar states in a field as for J 2 = 0, and for 1/8 < J 2 /J 1 < 1 they select the canted stripe state for all non-zero fields. The transition between the two states is first order in all fields, with the hysteresis width set by quantum fluctuations. We study the model with arbitrary S, including S = 1/2, near the saturation field by exploring the fact that near saturation the density of bosons is small for all S.We show that for S > 1, the transition remains first order, with a finite hysteresis width, but for S = 1/2 and, possibly, S = 1, there appears a new intermediate phase without a quasi-classical long-range order.