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We initiate a study of (real or complex) Banach algebras associated totwisted \'etale groupoids $(\mathcal{G},\mathcal{L})$ and to twisted inversesemigroup actions, which provides a general unifying framework for numerousrecent papers on $L^p$-operator algebras and the fully developed theory ofgroupoid $C^*$-algebras. Our main result is a version of the disintegrationtheorem that gives a bijective correspondence between representations of thetwisted groupoid Banach algebra $F(\mathcal{G},\mathcal{L})$ and covariantrepresentations of an inverse semigroup action. This powerful tool allows us tostudy $F(\mathcal{G},\mathcal{L})$ in terms of representations of the Banachalgebra $C_0(X)$ and a (twisted) inverse semigroup $S$ of partial isometriessubject to some relations. This works best when the groupoid is ample or whenthe target Banach algebra $B$ is a dual Banach algebra, for example when$B=B(E)$ for a reflexive Banach space $E$. When $E=L^p(\mu)$ for $p\in(1,\infty)\setminus \{2\}$ these relations force the partial isometries to bespatial, which explains why Phillips and others define $L^p$-analogues of Cuntzor graph algebras in terms of spatial partial isometries - the groupoid modelforces that.  We introduce and analyze in more detail $L^p$-operator algebras$F^{p}(\mathcal{G},\mathcal{L})$, $p\in[1,\infty]$, which are universal forrepresentations of $F(\mathcal{G},\mathcal{L})$ on $L^p$-spaces, and whichgeneralize algebras $F^{p}(\mathcal{G})$ introduced recently by Gardella andLupini. As more concrete examples we discuss Banach (and $L^p$-operatoralgebras) associated to twisted partial group actions, twisted Renault-Deaconugroupoids and directed graphs. We also introduce tight inverse semigroup Banachalgebras that cover all ample groupoid Banach algebras.

Banach algebras associated to twisted \'etale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces