The experimental discovery of correlated insulators and superconductivity in highly tunable Van der Waals
heterostructures, such as twisted bilayer graphene, has highlighted the role of moiré patterns, resulting from tiny relative
twists or lattice constant mismatches, in realizing strongly correlated physics. A key ingredient is the existence of very
narrow flat bands where interaction effects are dominant.

In this thesis and the accompanying papers, we theoretically study a number of experimentally relevant moiré systems. We generally show that strong interactions combined with the geometry and the topology of the underlying flat bands can result in a plethora of distinct quantum many-body phases ranging from topological order to multiferroicity. Of particular importance are lattice analogues of the fractional quantum Hall effect known as fractional Chern insulators. They harbour peculiar phenomena such as fractional charge and statistics and provide a route towards realizing topologically ordered states at high temperature. A ubiquitous feature of the many-body physics is the emergence of unique particle-hole dualities driven by the geometry of band-projected interactions.