  We present a new approach to graph limit theory which unifies and generalizes
the two most well developed directions, namely dense graph limits (even the
more general $L^p$ limits) and Benjamini--Schramm limits (even in the stronger
local-global setting). We illustrate by examples that this new framework
provides a rich limit theory with natural limit objects for graphs of
intermediate density. Moreover, it provides a limit theory for bounded
operators (called $P$-operators) of the form $L^\infty(\Omega)\to L^1(\Omega)$
for probability spaces $\Omega$. We introduce a metric to compare $P$-operators
(for example finite matrices) even if they act on different spaces. We prove a
compactness result which implies that in appropriate norms, limits of uniformly
bounded $P$-operators can again be represented by $P$-operators. We show that
limits of operators representing graphs are self-adjoint, positivity-preserving
$P$-operators called graphops. Graphons, $L^p$ graphons and graphings (known
from graph limit theory) are special examples for graphops. We describe a new
point of view on random matrix theory using our operator limit framework.
