The Cosmological Constant Problem: Why it's hard to get Dark Energy from Micro-physics

These notes present a brief introduction to `naturalness' problems in cosmology, and to the Cosmological Constant Problem in particular. The main focus is the `old' cosmological constant problem, though the more recent variants are also briefly discussed. Several notions of naturalness are defined, including the closely related ideas of technical naturalness and `t Hooft naturalness, and it is shown why these naturally arise when cosmology is embedded within a framework --- effective field theories --- that efficiently captures what is consistent with what is known about the physics of smaller distances. Some care is taken to clarify conceptual issues, such as the relevance or not of quadratic divergences, about which some confusion has arisen over the years. A set of minimal criteria are formulated against which proposed solutions to the problem can be judged, and a brief overview made of the general limitations of most of the approaches. A somewhat more in-depth discussion is provided of what I view as the most promising approach. These notes are aimed at graduate students with a basic working knowledge of quantum field theory and cosmology, but with no detailed knowledge of particle physics.