This thesis makes contributions to extremal combinatorics, specifically extremal set theory questions and their analogs in other structures. Extremal set theory studies how large or small a family of subsets of a finite set $X$ can be under various constraints. By replacing the set $X$ with another finite object, one can pose similar questions about families of other structures. Remarkably, a question and its analogs essentially have the same answer, regardless of the object. Despite these similarities, not much is known about analogs because standard techniques do not always apply. Our main results establish analogs of extremal set theory results for structures such as vector spaces and subsums of a finite sum. We also study intersecting families and shadows in their classical context of sets by researching a conjecture of Frankl and F\"{u}redi."