  We review the most recent progress in our understanding of quantum mechanical
observables in cosmology in the perturbative regime. It relies on an approach
that considers them directly as functions of the data at the space-like
boundary at future infinity prescinding from the explicit time evolution. It
takes inspiration from the on-shell formulation of perturbative scattering
amplitudes developed in the past 20 years: starting with the requirement of
consistency with some fundamental principles such as causality, unitarity and
locality, it provides different ways of phrasing and extracting predictions. In
this review, we aim to provide a pedagogical treatment of the most recent
insights about the analytic structure of the perturbative quantum mechanical
observables in cosmology, its relation to fundamental principles as well as
physical processes, and how such observables and their features emerge from
novel well-defined mathematical objects with their own first principle
definition. The review is divided in three parts: Part 0 discusses the
definition of quantum mechanical observables in cosmology and some general
principles; Part I reviews the boundary approach to the analysis and
computation of the perturbative wavefunction of the universe; Part II provides
an introduction to the combinatorial-geometrical description of cosmological
processes in terms of cosmological polytopes.
