  In four space-time dimensions, there exists a special infinite-parameter
family of chiral modified gravity theories. They are most properly described by
a connection field, with space-time metric being a secondary and derived
concept. All these theories have the same number of degrees of freedom as
general relativity, which is the only parity-invariant member of this family.
Modifications of general relativity can be arranged so as to become important
in regions with large curvature. In this paper we review how a certain simple
modification of this sort can resolve the Schwarzschild black-hole and Kasner
anisotropic singularities of general relativity. In the corresponding
solutions, the fundamental connection field is regular in space-time.
