  The long-range gravitational terms associated with tidal forces,
frame-dragging effects, and gravitational waves are described by the Weyl
conformal tensor, the traceless part of the Riemann curvature that is not
locally affected by the matter field. The Ricci and Bianchi identities provide
a set of dynamical and kinematic equations governing the matter coupling and
evolution of the electric and magnetic parts of the Weyl tensor, so-called
gravitoelectric and gravitomagnetic fields. A detailed analysis of the Weyl
gravitoelectromagnetic fields can be conducted using a number of algebraic and
differential identities prescribed by the 1+3 covariant formalism. In this
review, we consider the dynamical constraints and propagation equations of the
gravitoelectric/-magnetic fields and covariantly debate their analytic
properties. We discuss the special conditions under which gravitational waves
can propagate, the inconsistency of a Newtonian-like model without
gravitomagnetism, the nonlinear generalization to multi-fluid models with
different matter species, as well as observational effects caused by the Weyl
fields via the kinematic quantities. The 1+3 tetrad and 1+1+2 semi-covariant
methods, which can equally be used for gravitoelectromagnetism, are briefly
explained, along with their correspondence with the covariant formulations.
