Reaction-diffusion equations with delay of the form \[ {\partial u\over\partial t} (t,x)= \alpha\Delta u(t,x)+ \beta u(t,x)-\gamma u(t,x)f(u_t(\cdot,x)), \]
\[ u= 0\quad\text{on }[0,\infty)\times\partial\Omega,\quad u_0=\phi\quad\text{in }[-r,0]\times\Omega, \] are considered. Existence, uniqueness, and regularity are proved in standard ways. For asymptotic behaviour, the results are expressed in terms of the first eigenvalue \(\lambda_1\) of the Dirichlet problem associated with the operator \(-\Delta\). In fact, if \(\beta\leq\alpha\lambda_1\) then the zero solution is globally attracting, while if \(\beta>\alpha\lambda_1\) there is a nontrivial steady-state solution.