This paper presents a backstepping approach to design controllers for different components in hybrid AC/DC microgrids. The DC-side of the proposed microgrid structure includes a solar photovoltaic (PV) unit with a DC-DC boost converter, a permanent magnet synchronous generator (PMSG)-based wind generator with a rectifier, a battery energy storage system (BESS) with a bidirectional DC-DC converter, and DC loads. The AC part incorporates a synchronous generator along with AC loads while a bidirectional voltage source converter (VSC) is used between the DC and AC buses for transferring power from one side to another. The detailed dynamical models of all these components (except loads) are used to design the controllers using the proposed nonlinear backstepping approach. The controllers are designed in a decentralized way where the main objectives are to ensure appropriate power sharing while maintaining desired voltages at the common buses on both sides. The stability of the hybrid AC/DC microgrid is theoretically analyzed through control Lyapunov functions using the controllers obtained from the proposed control scheme. Simulation studies are carried out to demonstrate the performance of the proposed scheme under different operating scenarios as compared to a nonlinear sliding mode control scheme.