Energy infrastructures from different sectors, such as power networks and natural gas networks, are traditionally planned and operated dependently, necessitating sufficient coordination. The lack of coordination results in low energy efficiency and high operating costs. Meanwhile, market competitions and natural disasters might disrupt the coordination of such energy system. In this chapter, the basics and challenges for implementing integrated power and gas systems (IPGSs) are presented. Additionally, recent studies that have proposed methods to maintain the coordination and safe operation of the IPGS in energy market environment and mitigate catastrophic events damage are compared and analyzed. Moreover, the base operation problem of IPGS is presented, which is commonly used in the literature. Additionally, convexification methods including second-order cone relaxation and piecewise linearization methods are explained. Overall, this chapter presents an overview on the progress and challenges for implementing IPGS in market and enhancing its resilience. Finally, the basic optimization problem for the operation of IPGS is implemented in GAMS, and the numerical results with a market perspective are presented to effectively demonstrate the interdependencies of such energy systems which highlights the importance of coordination.