The energy sector is currently undergoing a rapid transformation with the integration of power electronic converter (PEC)-interfaced renewable energy sources (RES), such as wind and solar photovoltaic (PV) systems, at both the transmission and distribution networks. Power system stability has been significantly influenced by this power grid transformation. This paper comprehensively reviews major power system stability issues affected due to large-scale integration of PEC-interfaced RES in power grids, with some example case studies relevant for each stability category. According to the review, stability issues are mainly originating from reduction in synchronous inertia, reduction in reactive power reserve, low short-circuit strength of the power network, and fault ride-through (FRT) strategy/capability of the PEC-interfaced RES. Decrease in synchronous inertia could affect both the rotor angle stability and the frequency stability, while decrease in short-circuit strength and reactive power reserve could cause voltage stability and rotor angle stability issues in power networks. Sub-synchronous control interactions are also receiving a lot of attention by the power industry due to increasing oscillatory stability incidents reported in power networks with PEC-interfaced RES. FRT capabilities/strategies of PEC-interfaced RES are also playing a pivotal role in power grid stability due to its influence on active and reactive power, hence more emphasis should be placed on FRT schemes of PEC-interfaced RES, since future power grids are expected to operate with 100% PEC-interfaced generation sources. Stability improvement strategies could be implemented to address multiple stability issues in PEC-interfaced power networks; however, rigorous stability studies are required to identify the optimal conditions to implement these improvement strategies. Furthermore, ongoing structural changes in power grids to accommodate remotely sited PEC-interfaced RES are also influencing the stability of power grids. Therefore, all these factors must be carefully considered by system operators when planning and operating power grids in a secure and stable manner with high penetration levels of PEC-interfaced RES.