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A very useful idea in knot theory, dating back to at least the 1930's, is to study knotted curves in space by looking at certain types of embedded surfaces associated to them. In this talk, we will look at two kinds of such surfaces: Seifert surfaces and bridge spheres. After describing some numerical invariants of knots which arise from these surfaces, I will discuss recent work of mine (http://arxiv.org/abs/1602.06051) which leads to a new relationship between two of them. The key tool involved, which I'll informally describe, is a new method of using bridge spheres to decompose Seifert surfaces in a special way. No prior knowledge of knot theory will be assumed.