We use the Nernst effect to delineate the boundary of the pseudogap phase in the temperature-doping phase diagram of hole-doped cuprate superconductors. New data for the Nernst coefficient ν(T) of YBa2Cu3Oy (YBCO), La1.8-xEu0.2SrxCuO4 (Eu-LSCO), and La1.6-xNd0.4SrxCuO4 (Nd-LSCO) are presented and compared with previously published data on YBCO, Eu-LSCO, Nd-LSCO, and La2-xSrxCuO4 (LSCO). The temperature Tν at which ν/T deviates from its high-temperature linear behavior is found to coincide with the temperature at which the resistivity ρ(T) deviates from its linear-T dependence, which we take as the definition of the pseudogap temperature T - in agreement with the temperature at which the antinodal spectral gap detected in angle-resolved photoemission spectroscopy (ARPES) opens. We track T as a function of doping and find that it decreases linearly vs p in all four materials, having the same value in the three LSCO-based cuprates, irrespective of their different crystal structures. At low p,T is higher than the onset temperature of the various orders observed in underdoped cuprates, suggesting that these orders are secondary instabilities of the pseudogap phase. A linear extrapolation of T (p) to p=0 yields T (p→0)≃TN(0), the Néel temperature for the onset of antiferromagnetic order at p=0, suggesting that there is a link between pseudogap and antiferromagnetism. With increasing p,T (p) extrapolates linearly to zero at p≃pc2, the critical doping below which superconductivity emerges at high doping, suggesting that the conditions which favor pseudogap formation also favor pairing. We also use the Nernst effect to investigate how far superconducting fluctuations extend above the critical temperature Tc, as a function of doping, and find that a narrow fluctuation regime tracks Tc, and not T. This confirms that the pseudogap phase is not a form of precursor superconductivity, and fluctuations in the phase of the superconducting order parameter are not what causes Tc to fall on the underdoped side of the Tc dome.