  In this article we give an expository account of the holomorphic motion
theorem based on work of M\`a\~n\'e-Sad-Sullivan, Bers-Royden, and Chirka.
After proving this theorem, we show that tangent vectors to holomorphic motions
have $|\epsilon \log \epsilon|$ moduli of continuity and then show how this
type of continuity for tangent vectors can be combined with Schwarz's lemma and
integration over the holomorphic variable to produce H\"older continuity on the
mappings. We also prove, by using holomorphic motions, that Kobayashi's and
Teichm\"uller's metrics on the Teichm\"uller space of a Riemann surface
coincide. Finally, we present an application of holomorphic motions to complex
dynamics, that is, we prove the Fatou linearization theorem for parabolic germs
by involving holomorphic motions.
