There are many natural situations where a group with some additional structure admits an infinitary operation, i.e. an infinite sum, product, or composition. Fundamental groups and groupoids of topological spaces with non-trivial local structure (e.g. the Hawaiian earring, Menger Curve, and Sierpinski Carpet) provide natural models of algebraic structures with non-commutative infinite product operations. Since the 1990’s, significant progress has been made in the development and application of these topological-algebraic objects, culminating in Katsuya Eda’s remarkable homotopy classification of one-dimensional Peano continua. In this talk, I’ll give an introduction to this area and discuss characterizations and problems related to the well-definedness of infinitary operations on homotopy classes.