  This paper is a pedagogical yet critical introduction to the quantum
description of unstable systems, mostly at the level of a graduate quantum
mechanics course. Quantum decays appear in many different fields of physics,
and their description beyond the exponential approximation is the source of
technical and conceptual challenges. In this article, we present both general
methods that can be adapted to a large class of problems, and specific
elementary models to describe phenomena like photo-emission, beta emission and
tunneling-induced decays. We pay particular attention to the emergence of
exponential decay; we analyze the approximations that justify it, and we
present criteria for its breakdown. We also present a detailed model for
non-exponential decays due to resonance, and an elementary model describing
decays in terms of particle-detection probabilities. We argue that the
traditional methods for treating decays face significant problems outside the
regime of exponential decay, and that the exploration of novel regimes of
current interest requires new tools.
