  A theory of $\infty$-properads is developed, extending both the Joyal-Lurie
$\infty$-categories and the Cisinski-Moerdijk-Weiss $\infty$-operads. Every
connected wheel-free graph generates a properad, giving rise to the graphical
category $\Gamma$ of properads. Using graphical analogs of coface maps and the
properadic nerve functor, an $\infty$-properad is defined as an object in the
graphical set category $Set^{\Gamma^{op}}$ that satisfies some inner horn
extension property. Symmetric monoidal closed structures are constructed in the
categories of properads and of graphical sets. Strict $\infty$-properads, in
which inner horns have unique fillers, are given two alternative
characterizations, one in terms of graphical analogs of the Segal maps, and the
other as images of the properadic nerve. The fundamental properad of an
$\infty$-properad is characterized in terms of homotopy classes of
$1$-dimensional elements. Using all connected graphs instead of connected
wheel-free graphs, a parallel theory of $\infty$-wheeled properads is also
developed.
