Here's a new paper that studies categorified groups and Lie groups: Higher-Dimensional Algebra V: 2-Groups John C. Baez and Aaron D. Lauda Abstract: A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G x G -> G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x tensor y and y tensor x are isomorphic to 1. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse x* and isomorphisms i_x: 1 -> x tensor x*, e_x: x* tensor x -> 1 forming an adjunction. We define 2-categories of weak and coherent 2-groups, construct an "improvement" 2-functor which turns weak 2-groups into coherent ones, and prove this 2-functor is a 2-equivalence of 2-categories. We also internalize the concept of coherent 2-group, which gives a way to define topological 2-groups, Lie 2-groups, affine 2-group schemes, and the like. We conclude with a tour of examples. Diagrammatic methods are emphasized throughout - especially string diagrams. This paper will soon appear on the mathematics arXiv, but their computer seems unable to draw some of the pictures correctly, so I urge you to try this PDF version instead: http://math.ucr.edu/home/baez/hda5.pdf The next paper in this series will study categorified Lie algebras.