  We display methods that allow for computations of spectra, pseudospectra and
resolvents of linear operators on Hilbert spaces and also elements in unital
Banach algebras. The paper considers two different approaches, namely,
pseudospectral techniques and polynomial numerical hull theory. The former is
used for Hilbert space operators whereas the latter can handle the general case
of elements in a Banach algebra. This approach leads to multicentric
holomorphic calculus. We also discuss some new types of pseudospectra and the
recently defined Solvability Complexity Index
