Dark sectors, consisting of new, light, weakly-coupled particles that do not interact with the known strong, weak, or electromagnetic forces, are a particularly compelling possibility for new physics. Nature may contain numerous dark sectors, each with their own beautiful structure, distinct particles, and forces. This review summarizes the physics motivation for dark sectors and the exciting opportunities for experimental exploration. It is the summary of the Intensity Frontier subgroup "New, Light, Weakly-coupled Particles" of the Community Summer Study 2013 (Snowmass). We discuss axions, which solve the strong CP problem and are an excellent dark matter candidate, and their generalization to axion-like particles. We also review dark photons and other dark-sector particles, including sub-GeV dark matter, which are theoretically natural, provide for dark matter candidates or new dark matter interactions, and could resolve outstanding puzzles in particle and astro-particle physics. In many cases, the exploration of dark sectors can proceed with existing facilities and comparatively modest experiments. A rich, diverse, and low-cost experimental program has been identified that has the potential for one or more game-changing discoveries. These physics opportunities should be vigorously pursued in the US and elsewhere.