  A set of canonical parahermitian connections on an almost paraHermitian
manifold is defined. ParaHermitian version of the Apostolov-Gauduchon
generalization of the Goldberg-Sachs theorem in General Relativity is given. It
is proved that the Nijenhuis tensor of a Nearly paraK\"ahler manifolds is
parallel with respect to the canonical connection. Salamon's twistor
construction on quaternionic manifold is adapted to the paraquaternionic case.
A hyper-paracomplex structure is constructed on Kodaira-Thurston (properly
elliptic) surfaces as well as on the Inoe surfaces modeled on $Sol^4_1$. A
locally conformally flat hyper-paraK\"ahler (hypersymplectic) structure with
parallel Lee form on Kodaira-Thurston surfaces is obtained. Anti-self-dual
non-Weyl flat neutral metric on Inoe surfaces modeled on $Sol^4_1$ is
presented. An example of anti-self-dual neutral metric which is not locally
conformally hyper-paraK\"ahler is constructed.
