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In the last decade, the concept of grid-forming (GFM) converters has been introduced for microgrids and islanded power systems. Recently, the concept has been proposed for use in wider interconnected transmission networks, and several control structures have thus been developed, giving rise to discussions about the expected behaviour of such converters. In this paper, an overview of control schemes for GFM converters is provided. By identifying the main subsystems in respect to their functionalities, a generalized control structure is derived and different solutions for each of the main subsystems composing the controller are analyzed and compared. Subsequently, several selected open issues and challenges regarding GFM converters, i. e. angle stability, fault ride-through (FRT) capabilities, and transition from islanded to grid connected mode are discussed. Perspectives on challenges and future trends are lastly shared.