In this talk I will discuss the asymptotic stability problem for KdV solitons on the right half-line. Unlike standard KdV, these are not exact solutions to the equations posed on the half-line, and, contrary to NLS, no exact Soliton solution seems to exist. In a previous result, we showed that Solitons of the KdV equation posed in the entire line, placed sufficiently far from the origin, are stable in the half-line energy space, and assuming homogeneous boundary conditions. Now, we prove their asymptotic stability in the energy space, and provide decay properties for all remaining regions, except the “small soliton region". For the proof we follow the ideas by Martel and Merle for the big soliton part, and for the linearly dominated region we follow recent results on generalized KdV decay of Muñoz et al. This is a joint work with Claudio Muñoz (Universidad de Chile).