Given a vector bundle V with a linear connection, the holonomy of the connection is a group generated by parallel transport along closed loops. The most well-known case is when V is the tangent bundle of a (n-dimensional) Riemannian manifold, and the connection is the Levi-Civita connection, in which case the holonomy is a subgroup of the orthogonal group O(n). Two important results lie at the foundation of Riemannian holonomy theory: the De Rham Decomposition Theorem and Berger’s Theorem. The first result says that a Riemannian manifold whose holonomy group preserves a non-trivial subspace of \R^n splits, at least locally, into a product of Riemannian manifolds, and the holonomy group is isomorphic to the product of Riemannian holonomy groups. The second result gives a short list of the subgroups of O(n) which could occur as the holonomy group of a simply connected Riemannian manifold with no invariant subspaces of \R^n. A major accomplishment was to show that all the groups on Berger’s list do actually occur as the holonomy of some (compact) Riemannian manifold, and it took some 40 years until this was settled for the last remaining groups. If we no longer fix a Riemannian structure on a manifold, but only a conformal equivalence class of Riemannian metrics, then the tangent bundle no longer admits a canonical linear connection, but there is vector bundle of rank n+2 which does have a canonical linear connection coming from the Cartan connection of a conformal manifold. The holonomy group of this connection is a subgroup of the pseudo-orthogonal group O(n+1,1), and there are many open questions concerning the possibilities for these conformal holonomy groups. In this talk I will give an introduction to the main results which have been established recently. This includes a conformal analog of the De Rham Theorem, which relates holonomy-invariant subspaces of \R^{n+2} to the existence of a certain product of Einstein metrics in the conformal class. There has still been very little progress toward an analog of Berger's list for conformal holonomy, but I will review what is known.