Nonlocal    reaction-diffusion     equations    of     the   form    $
u_t=u_{xx}+F(u,a(u))$,  where $a(u)=\int_{-1}^1 u(x)dx$, are considered
together with Neumann or   Dirichlet boundary conditions. One  of  the
main results deals with  linearizations at equilibria. It  states that
for any  given  set of complex  numbers one  can arrange, choosing the
equation properly, that  this set is contained in  the spectrum of the
linearization. The second  main result  shows  that equations  of  the
above form can undergo a  supercritical Hopf bifurcation leading to an
asymptotically stable periodic solutions.