  We introduce Banach algebras associated to a twisted \'etale groupoid
$(\mathcal{G},\mathcal{L})$ and to twisted inverse semigroup actions. This
provides a unifying framework for numerous recent papers on $L^p$-operator
algebras and the theory of groupoid $C^*$-algebras. We prove a disintegration
theorem that allows us to study Banach algebras associated to
$(\mathcal{G},\mathcal{L})$ as universal Banach algebras generated by $C_0(X)$
and a twisted inverse semigroup $S$ of partial isometries subject to some
relations. Our theory works best when the groupoid is ample or when the target
of a representation is a dual Banach algebra $B$, so for instance $B=B(E)$
where $E$ is a reflexive Banach space. In particular, it implies that in the
constructions of $L^p$-analogues of Cuntz or graph algebras, by Phillips and
Corti\~{n}as--Rodr\'{\i}guez, the use of \emph{spatial partial isometries} is
not an assumption, in fact it is forced by the relations.
  We establish fundamental norm estimates and hierarchy for full and reduced
$L^p$-operator algebras for $(\mathcal{G},\mathcal{L})$ and $p \in [1,\infty]$,
whose special cases have been studied recently by Gardella--Lupini,
Choi--Gardella--Thiel and Hetland--Ortega. We also introduce tight inverse
semigroup Banach algebras that cover ample groupoid Banach algebras, and we
discuss Banach algebras associated to twisted partial group actions, twisted
Renault--Deaconu groupoids and directed graphs. Our Banach Cuntz algebras are
related to the construction of Daws--Horv\'{a}th.
  We aim for a high degree of generality throughout: our results cover
non-Hausdorff \'{e}tale groupoids and both real and complex algebras. Some of
our results seem to be absent from the literature even for $C^*$-algebras.
