The classical $\Delta$-system lemma is one of the foundational results of combinatorial set theory and is an important tool in many forcing arguments. $\Delta$-systems can be seen as inherently one-dimensional objects, though, so arguments about higher-dimensional phenomena often call for higher-dimensional generalizations of the classical $\Delta$-system lemma. Such generalizations were first developed and applied by Todorcevic and Shelah in the 1980s, and they have seen increased use in recent years. In this talk, we will present a particular definition of "higher-dimensional $\Delta$-system", isolate optimal hypotheses under which such generalized $\Delta$-systems necessarily exist, and present some applications to questions arising from Ramsey theory and homological algebra.