Several challenges are encountered while integrating microgrids (MGs) with an existing utility grid. These are low inertia, intermittent nature of renewable energy resources (RESs), sensor and actuator faults, unbalanced and nonlinear loads, supply-demand mismatch, and uncertain switching functions of the power electronic converter. MG control is fully dependent on the communication network, which is also susceptible to different types of failures such as noise in communication links, communication delay, limited bandwidth, packet dropout, and various malicious cyber attacks. Very few papers are available in the literature that focus on the review of control schemes for MG. Consequently, this paper presents a comprehensive review of different robust and adaptive control schemes that address the challenges encountered due to communication constraints, uncertainties, and disturbances for different MG topologies such as AC, DC, and hybrid AC/DC MGs, as well as the current research trends in the field of robust and adaptive control. It can be concluded that to achieve control objectives of MG and overcome the existing challenges, robust and adaptive controllers show significantly improved performance in terms of transient and steady-state behavior and robustness as compared to traditional controllers.