It has long been appreciated that exactly solved mathematical models describing the statistical mechanics of interacting particles have played a key role in the development of formerly unrelated areas of mathematics and theoretical physics, such as the study of knots, links and braids, quantum groups, combinatorics, conformal field theory and condensed matter physics. However, over the past few years striking experimental achievements in trapping and cooling atoms in one-dimensional optical waveguides have provided remarkable realisations of exactly solved models in the lab. More generally the study of cold atomic matter provides a unique environment to explore novel quantum many-body effects like quantum liquids, quantum correlations and quantum criticality. In this talk I will describe some of these fundamental mathematical models and their relevance to recent and future experiments on such exotic many-body physics.