A central topic in condensed matter research during the last decades has been the study and classification of topological
phases of matter. Topological insulators in particular, a subset of symmetry protected topological phases, have been investigated for over a decade. In recent years, several extensions to this formalism have been proposed to study more unconventional systems. In this thesis we explore two of these extensions, where key assumptions in the original formalism are removed. The first case is critical systems, which have no energy gap. Conventional topological invariants are discontinuous at topological transitions, and therefore not well-defined for critical systems. We propose a method for generalizing conventional topological invariants to critical systems and show robustness to disorder that preserves
criticality.

The second case involves non-Hermitian systems, which appear in effective descriptions of dissipation, where we
study the entanglement spectrum and its connection to topological invariants. Furthermore, by introducing non-Hermiticity to critical systems we show how the winding numbers that characterize some topological phases of the non-Hermitian
system, as well as topological signatures in the entanglement spectrum, can be obtained from the related critical model.