  A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the
authors and shown to be an integrable infinite dimensional Hamiltonian
equation. Unlike the classical (local) case, here the nonlinearly induced
"potential" is $PT$ symmetric thus the nonlocal NLS equation is also $PT$
symmetric. In this paper, new {\it reverse space-time} and {\it reverse time}
nonlocal nonlinear integrable equations are introduced. They arise from
remarkably simple symmetry reductions of general AKNS scattering problems where
the nonlocality appears in both space and time or time alone. They are
integrable infinite dimensional Hamiltonian dynamical systems. These include
the reverse space-time, and in some cases reverse time, nonlocal nonlinear
Schr\"odinger, modified Korteweg-deVries (mKdV), sine-Gordon, $(1+1)$ and
$(2+1)$ dimensional three-wave interaction, derivative NLS, "loop soliton",
Davey-Stewartson (DS), partially $PT$ symmetric DS and partially reverse
space-time DS equations. Linear Lax pairs, an infinite number of conservation
laws, inverse scattering transforms are discussed and one soliton solutions are
found. Integrable reverse space-time and reverse time nonlocal discrete
nonlinear Schr\"odinger type equations are also introduced along with few
conserved quantities. Finally, nonlocal Painlev\'e type equations are derived
from the reverse space-time and reverse time nonlocal NLS equations.
