  Soft materials, such as colloidal suspensions, polymer solutions, and
biological systems, are typically multicomponent mixtures of macromolecules and
simpler components (e.g., microions, monomers, solvent) that can assemble into
complex structures spanning broad length and time scales. Many characteristic
traits of soft matter, e.g., mechanical fragility, sensitivity to external
influence, and tunable thermal and optical properties, emerge from a mingling
of microscopic and mesoscopic constituents. Large asymmetries in size and
charge between macromolecules and microscopic components often make impractical
the explicit modeling of all degrees of freedom over physically significant
length and time scales. The complexity often can be reduced, however, by
pre-averaging (coarse-graining) the degrees of freedom of some microscopic
components, thus mapping the real system onto a simpler model of fewer
components, but governed by effective interparticle interactions. This chapter
reviews the statistical mechanical foundations of effective interactions in
soft materials and discusses illustrative applications to charged colloids,
polyelectrolytes, and colloid-polymer mixtures
