Earlier studies on network robustness have mainly focused on the integrity of functional components such as the giant connected component in a network. Generalized k-core (Gk-core) has been recently investigated as a core structure obtained via a k-leaf removal procedure extending the well-known leaf removal algorithm. Here, we study analytically and numerically the network robustness in terms of the numbers of nodes and edges in Gk-core against random attacks (RA), localized attacks (LA) and targeted attacks (TA), respectively. In addition, we introduce the concept of Gk-core stability to quantify the extent to which the Gk-core of a network contains the same nodes under independent multiple RA, LA and TA, respectively. The relationship between Gk-core robustness and stability has been studied under our developed percolation framework, which is of significance in better understanding and design of resilient networks.