Two-dimensional moiré superlattices have emerged as an ideal system to study the many-body interactions and correlated states. Quantum many-body problems are difficult to solve because the dimensions of Hilbert space grow exponentially with the size of a quantum system. Only a limited number of quantum models have been solved exactly. In this regard, quantum simulators are of significant importance. Moiré superlattices serve as ideal quantum simulators due to their high tunability of parameters. The filling factor of the moiré energy band can be changed by varying the gate voltage, while other electronic properties can be modified by easily accessible external factors such as an external electric or magnetic fields, strain or a twist angle between the atomic layers forming a moiré pattern. Various many-body interacting states have already been engineered, such as Mott insulator states at one hole per superlattice unit cell, superconductivity, generalized Wigner crystal states, different charge orders or fractional Chern insulators at partial fillings of topologically non-trivial energy bands. During my talk I will review recent experimental and theoretical results in this field.