  We investigate models of interacting dark matter and dark energy for the
universe in a spatially flat Friedmann-Robertson-Walker (FRW) space-time. We
find the "source equation" for the total energy density and determine the
energy density of each dark component. We introduce an effective one-fluid
description to evidence that interacting and unified models are related with
each other, analyze the effective model and obtain the attractor solutions. We
study linear and nonlinear interactions, the former comprises a linear
combination of the dark matter and dark energy densities, their first
derivatives, the total energy density, its first and second derivatives and a
function of the scale factor. The latter is a possible generalization of the
linear interaction consisting of an aggregate of the above linear combination
and a significant nonlinear term built with a rational function of the dark
matter and dark energy densities homogeneous of degree one. We solve the
evolution equations of the dark components for both interactions and examine
exhaustively several examples. There exist cases where the effective one-fluid
description produces different alternatives to the $\La$CDM model and cases
where the problem of coincidence is alleviated. In addition, we find that some
nonlinear interactions yield an effective one-fluid model with a Chaplygin gas
equation of state, whereas others generate cosmological models with de Sitter
and power-law expansions. We show that a generic nonlinear interaction induces
an effective equation of state which depends on the scale factor in the same
way that the variable modified Chaplygin gas model, giving rise to the "relaxed
Chaplygin gas model".
