  We discuss some properties of recently proposed models of a ghost-free
gravity. For this purpose we study solutions of linearized gravitational
equations in the framework of such a theory. We mainly focus on the version of
the ghost-free theory with the exponential modification
$\exp(\Box/\mu^2)\Box^{-1}$ of the free propagator. The following three
problems are discussed: (i) Gravitational field of a point mass; (ii) Penrose
limit of a point source boosted to the speed of light; and (iii) Spherical
gravitational collapse of null fluid. For the first problem we demonstrate that
it can be solved by using the method of heat kernels and obtain a solution in a
spacetime with arbitrary number of dimensions. For the second problem we also
find the corresponding gyraton-type solutions of the ghost-free gravitational
equations for any number of dimensions. For the third problem we obtain
solutions for the gravitational field for the collapse of both "thin" and
"thick" spherical null shells. We demonstrate how the ghost-free modification
of the gravitational equations regularize the solutions of the linearized
Einstein equations and smooth out their singularities.
