ABSTRACT: I will discuss combinatorial methods for studying the topology of poset order complexes, focusing on the example of the partition lattice and its $GL_n(q)$-analogue. Over the past 20 to 30 years, a combinatorial technique called lexicographic shellability has been used to show that remarkably many combinatorial simplicial complexes have the homotopy type of a wedge of spheres, and in the process to compute coefficients in inclusion-exclusion counting formulas. The partition lattice is a particularly nice example, as well as an important one. About 10 years ago, Robin Forman introduced a new technique to combinatorics called discrete Morse theory, which may be applied to simplicial complexes and cell complexes of much more general topological type. After reviewing this theory, I will very briefly discuss machinery, much of which is joint work with Eric Babson and with Volkmar Welker, for applying discrete Morse theory to poset order complexes. Finally, I will describe a very recent application, developed jointly with Phil Hanlon and John Shareshian, to a poset whose GL_n(q)-representation on top homology seems to be the natural analogue for the S_n-representation on the top homology of the partition lattice.