Let k be a field and S = k [x1 , . . . , xn ] a polynomial ring. This thesis considers the structure of minmial free resolutions of monomial ideals in S . In Chapter 3 we study reverse lex ideals, and compare their properties to those of lex ideals. In particular we provide an analogue of Green's Theorem for reverse lex ideals. We also compare the Betti numbers of strongly stable and square-free strongly stable monomial ideals to those of reverse lex ideals. In Chapter 5 we study the minimal free resolution of the edge ideal of the complement of the n-cycle for n [GREATER-THAN OR EQUAL TO] 4 and construct a regular cellular complex which supports this resolution.