This report first summarizes the works about nonlinear conic optimization problems, especially about strong regularity and isolated calmness for nonlinear programming, second-order conic optimization and semidefinite programming. After that it is devoted to studying the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the constraint non-degeneracy and the second order sufficient condition are both needed for the KKT solution mapping to have the Aubin property.