Galileons are a class of scalar field theories which have been found to arise in a disparate variety of contexts and exhibit a host of interesting properties by themselves, both classical and quantum. They obey non-trivial shift symmetries which restrict their self-interactions to be of higher derivative form, yet their equations of motion remain second order so that they are free of ghost instabilities. Further, when used as a force mediator between massive objects, galileons provide a natural realization of the Vainshtein screening mechanism which shuts off the fifth force at distances close to massive sources. As such, they are well suited for cosmology and are naturally incorporated into theories of modified gravity such as the Dvali-Gabadadze-Porrati braneworld model and the de Rham-Gabadadze-Tolley theory of massive gravity. Treated as a quantum field theory, galileons obey a non-trivial non-renormalization theorem which proves that they are not renormalized to any numbers of loops. In this thesis, we explore the properties of galileon theories and their generalizations through a combination of geometric and algebraic means. On the geometry side, we demonstrate that generic galileon theories are naturally thought of as the description of branes moving in higher dimensional spacetimes. On the algebraic side, we show that there exists a precise interpretation in which galileons can be thought of as Goldstone modes which arise when spacetime symmetries are spontaneously broken. In particular, when viewed in this light the galileons are the analogue of the Wess-Zumino-Witten term of the chiral lagrangian and thus represent interactions which are technically special. These methods provide both new technical tools for analyzing galileon-like theories and offer conceptual changes for how these theories can be viewed.