  We develop an $\infty$-categorical version of the classical theory of
polynomial and analytic functors, initial algebras, and free monads. Using this
machinery, we provide a new model for $\infty$-operads, namely $\infty$-operads
as analytic monads. We justify this definition by proving that the
$\infty$-category of analytic monads is equivalent to that of dendroidal Segal
spaces, known to be equivalent to the other existing models for
$\infty$-operads.
