Microgrids are relatively smaller but complete power systems. They incorporate the most innovative technologies in the energy sector, including distributed generation sources and power converters with modern control strategies. In the future smart grids, they will be an essential element in their architecture. Their potential to offer many economic, social and environmental services through advanced electrical techniques has led to a growing interest in the theme. Although the islanding condition is a very important feature of microgrids, only with the implementation of grid connection and seamless transition they will demonstrate their full capacity. However, there are still many questions surrounding these operation modes and this paper tries to answer part of them. To do that, several aspects in the field are approached. The history and late development of microgrids are revisited. The main concepts are presented. The islanded mode is revised, since it is intrinsically linked to the other working states of the microgrid. The requirements for the interconnection of microgrids to an external grid are discussed. The operation elements are also analyzed. A crucial part of the grid-connected microgrids and their seamless transfer conditions, the control methods found in the literature are extensively reviewed. The paper is concentrated in the analysis of control methods for AC microgrids and AC power systems, therefore, it does not enter in detail or investigates profoundly the topologies applied in the power electronics structures nor DC microgrids and DC power systems.INDEX TERMS Microgrid, grid-connected mode, seamless transition, islanded mode, droop control, distributed generation, smart grid, power electronics.