The initial value problem (IVP) for the Korteweg-deVries (KdV) equation ∂ tu+u∂ xu+∂ 3 xu=0, u(x, 0) = φ(x) for x ∈ R, t ∈ R establishes a nonlinear map K from the space H s ( R) to the space C([− T, T]; H s ( R)). It has been known for many years that this map K is continuous [Bona and Smith (1975), Kato (1983)] and it was proved recently that this map is Lipschitz continuous [Kenig et al. (1993)]. It is shown in this paper that the nonlinear map K is infinitely many times Fréchet differentiable from H s ( R) to C([− T, T]; H s ( R)). Furthermore, it is proved that K has a Taylor series expansion at any given φ ∈ H s ( R), i.e., [formula] where K n(φ), the nth derivative of K at φ, is an n-linear map from the n-fold space of H s ( R) to C([− T, T]; H s ( R)) and the series converges in the space C([− T, T]; H s ( R)) uniformly for || h|| ≤ δ for some δ > 0. Each term y n = K n (φ)[ h n ] in the series solves a linearized KdV equation. Thus any solutions of the IVP (∗) can be obtained by solving a series of linear problems. By contrast, the corresponding map K p established by the initial value problem for the periodic KdV equation ∂ tu+u∂ xu+ ∂ 3 xu = 0, u(x, 0) = φ(x) for x ∈ S, t ∈ R, where S is the unit length circle in the plane, is known to be continuous only from H s ( S) to C([− T, T]; H s ( S)). This is due to the lack of smoothing effects for solutions of the periodic KdV equation. In this paper, it is shown that K p is Lipschitz continuous from H s+1 ( S) to C([− T, T]; H s ( S)) and is n times Fréchet differentiable from H s+ n+1 ( S) to C([− T, T]; H s ( S)) for any n ≥ 1. The method developed in this paper applies to other nonlinear dispersive wave equations.