  We treat spectral problems by twisted groupoid methods. To Hausdorff locally
compact groupoids endowed with a continuous $2$-cocycle one associates the
reduced twisted groupoid $C^*$-algebra. Elements (or multipliers) of this
algebra admit natural Hilbert space representations. We show the relevance of
the orbit closure structure of the unit space of the groupoid in dealing with
spectra, norms, numerical ranges and $\epsilon$-pseudospectra of the resulting
operators. As an example, we treat a class of pseudo-differential operators
introduced recently, associated to group actions. We also prove a Decomposition
Principle for bounded operators connected to groupoids, showing that several
relevant spectral quantities of these operators coincide with those of certain
non-invariant restrictions. This is applied to Toeplitz-like operators with
variable coefficients and to band dominated operators on discrete metric
spaces.
