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The initial motivation of Drinfeld and Jimbo for introducing quantum groups was to develop an algebraic framework for constructing solutions of the Yang-Baxter equation. Quantum affine algebras were important examples because their finite-dimensional representations give the so-called trigonometric solutions of the Yang-Baxter equation with spectral parameter. In this talk, I will try to explain some of the main features of this interesting class of representations, and how they resemble/differ from the more familar finite-dimensional representations of a complex semisimple Lie algebra. This will provide some background for my talk in the "Algebra and Discrete Mathematics" seminar, later in the afternoon.