We use the Nernst effect to delineate the boundary of the pseudogap phase in the temperature-doping phase diagram of cuprate superconductors. New data for the Nernst coefficient $ u(T)$ of YBa$_{2}$Cu$_{3}$O$_{y}$ (YBCO), La$_{1.8-x}$Eu$_{0.2}$Sr$_x$CuO$_4$ (Eu-LSCO) and La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (Nd-LSCO) are presented and compared with previous data including La$_{2-x}$Sr$_x$CuO$_4$ (LSCO). The temperature $T_ u$ at which $ u/T$ deviates from its high-temperature behaviour is found to coincide with the temperature at which the resistivity deviates from its linear-$T$ dependence, which we take as the definition of the pseudogap temperature $T^star$- in agreement with gap opening detected in ARPES data. We track $T^star$ 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 $p$, $T^star$ 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 $T^star(p)$ to $p=0$ yields $T^star(pto 0)simeq T_N(0)$, the Neel temperature for the onset of antiferromagnetic order at $p=0$, suggesting that there is a link between pseudogap and antiferromagnetism. With increasing $p$, $T^star(p)$ extrapolates linearly to zero at $psimeq p_{rm c2}$, the critical doping below which superconductivity emerges at high doping, suggesting that the conditions which favour pseudogap formation also favour pairing. We also use the Nernst effect to investigate how far superconducting fluctuations extend above $T_{rm c}$, as a function of doping, and find that a narrow fluctuation regime tracks $T_{rm c}$, and not $T^star$. This confirms that the pseudogap phase is not a form of precursor superconductivity.
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