  For a number of nonlocal nonlinear equations such as nonlocal, nonlinear
Schr\"odinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal,
saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL and
coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of
Jacobi elliptic functions as well as the corresponding hyperbolic soliton
solutions. Remarkably, in all the six cases, we find that unlike the
corresponding local cases, all the nonlocal models simultaneously admit both
the bright and the dark soliton solutions. Further, in all the six cases, not
only $\dn(x,m)$ and $\cn(x,m)$ but even their linear superposition is shown to
be an exact solution. Finally, we show that the coupled nonlocal NLSE not only
admits solutions in terms of Lam\'e polynomials of order 1, but it also admits
solutions in terms of Lam\'e polynomials of order 2, even though they are not
the solutions of the uncoupled nonlocal problem. We also remark on the possible
integrability in certain cases.
