We study the well-posedness of the initial value problem for fully nonlinear evolution equations, [ image ] where f may depend on up to the first three spatial derivatives of u. We make three primary assumptions about the form of [ image ] a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion. Because the third derivative of u is present in the right-hand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdV-type. We prove the well-posedness of the initial value problem in the Sobolev space [ image ], which is close to optimal for the energy estimates we make. The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data. There is evidence that the backward diffusion condition we express is optimal.