The fundamental group of a graph of groups is a concept thatgeneralizes both amalgamated free products and HNN extensions, twofundamental constructions in geometric group theory. Understanding thestructure of these constructions based on the structure of the componentgroups is an important topic. The centralizer of an element $gin G$ is the subgroup $Z_{G}(g) = { g'in G mid gg'=g'g}$, the set of all elements which commute with $g$. In this talk, I will introduce both algebraic methods (based on normal forms of elements) and geometric methods (based on actions ofthe group on graph theoretic trees) for computing the centralizers in thecases of amalgamated free products and HNN extensions.I also hope to indicate what we might expect to happen in the generalcase of fundamental groups of graphs of groups and to talk about somepossible generalizations of my work.