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K-theory, on the category of compact Hausdorff spaces, has been useful in many areas--topology, analysis, physics, etc. Because of this, different ways of representing elements of K*(X) have been developed. In this talk, we will introduce K-theory and discuss its use in some unexpectd settings. Then we will consider its representation in terms of families of unbounded self-adjoint Fredholm operators. This leads to the notions of spectral flow and the "index gerbe", which realize the first two components of the Chern character of the family. It turns out that the multiplicity of the spectrum of the operators in the family plays a role in studying these and higher invariants. This will be discussed in the last part and is joint work with Ron Douglas. I intend to make good part of the talk an introductory exposition which won't require much background.