Vertex decomposition, introduced by Provan and Billera in 1980, is an inductive strategy for breaking down and understanding simplicial complexes. A simplicial complex that is vertex decomposable is shellable, hence Cohen--Macaulay. Through the Stanley--Reisner correspondence, vertex decompositions of simplicial complexes inform the study of squarefree monomial ideals, which we will be interested in studying in their capacity as the defining ideals of initial varieties of combinatorially-defined varieties. One limitation of this story is that a great deal of the combinatorial information defining a variety may be lost when one jumps in one fell swoop to its initial variety. A generalization of vertex decomposition, called geometric vertex decomposition, introduced by Knutson, Miller, and Yong in 2009, allows one to take smaller steps towards the initial variety, preserving more of the original combinatorial information along the way. In this talk, we will describe and give examples of geometric vertex decompositions in combinatorially-natural settings and discuss relationships among geometric vertex decomposition, various algebro-geometric invariants, and Frobenius splitting. We will end with some open problems. This talk will touch on joint work with Andrew Berget, Emanuela De Negri, Elisa Gorla, Jenna Rajchgot, Lisa Seccia, Mayada Shahada, and Anna Weigandt.