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I will introduce the notion of the graph known as the "curve complex" for a surface, and discuss some of its applications to 3-manifolds. We will then discuss the complex of surfaces, and its relationship to the powerful notion of "thin position" for 3-maniolds, as well as other complexes that can be constructed from a 3-manifold. I will try to avoid getting weighed down in technical details, and instead focus on intuitive understanding and interesting examples and constructions of 3-manifolds. This talk draws on multiple sources, including John Hempel's landmark 1997 paper, "3-manifolds as viewed from the curve complex", work of Abigail Thompson and Martin Scharlemann, as well as more recent work of Jesse Johnson.