  In this paper, by introducing the notion of ``\textit{distributive
constant}'' of a family of hypersurfaces with respect to a projective variety,
we prove a second main theorem in Nevanlinna theory for meromorphic mappings
with arbitrary families of hypersurfaces in projective varieties. Our second
main theorem generalizes and improves previous results for meromorphic mappings
with hypersurfaces, in particular for algebraically degenerate mappings and for
the families of hypersurfaces in subgeneral position. The analogous results for
the holomorphic curves with finite growth index from a complex disc into a
project variety, and for meromorphic mappings on a complete K\"{a}hler manifold
are also given. For the last aim, we will prove a Schmidt's subspace theorem
for an arbitrary families of homogeneous polynomials, which is the counterpart
in Number theory of our second main theorem. Our these results are
generalizations and improvements of all previous results.
