  We give several characterizations of when a complete first-order theory $T$
is monadically NIP, i.e. when expansions of $T$ by arbitrary unary predicates
do not have the independence property. The central characterization is a
condition on finite satisfiability of types. Other characterizations include
decompositions of models, the behavior of indiscernibles, and a forbidden
configuration. As an application, we prove non-structure results for hereditary
classes of finite substructures of non-monadically NIP models that eliminate
quantifiers.
