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.