We will discuss special regularity properties of solutions to the IVP associated to the k-generalized KdV equations. In [1] we show that for data u0 ∈ H^3/4+ (R)

whose restriction belongs to H^k ((b, ∞)) for some k ∈ Z+ and b ∈ R, the restriction of the corresponding solution u(·, t) belongs to H^k ((?, ∞)) for any ? ∈ R

and any t ∈ (0, T ). Thus, this type of regularity propagates with infinite speed to its left as time evolves. This kind of regularity can be extended to a general

class of nonlinear dispersive equations. Recently, in [2] we proved that the solution flow of the k-generalized KdV equation does not preserve other kind of regularities exhibited by the initial data u0. If time allows we will discuss propagation of regularity in solutions of related dispersive equations.