Recent advances in materials science have established Moiré materials as a new highly tunable and versatile form of quantum matter. When two dimensional atomic layers are brought into proximity, a tiny relative twist or a slight lattice mismatch produces Moiré patterns manifested in a superlattice structure with a lattice constant that is much larger than the lattice constants of the constituent layers. The new length scale has dramatic consequences for the underlying properties. A particular distinctive feature of Moiré materials is the emergence of nearly flat bands upon tuning external parameters such as the twist angle or the applied gate voltage. In a flat band, the kinetic energy is quenched, and interactions are enhanced bringing us to the realm of strongly correlated systems. A prime example of Moiré materials is twisted bilayer graphene, formed by taking two graphene layers and twisting them relative to each other.

On the other hand, a famous class of interaction-induced phases of matter are fractional quantum Hall states and their lattice analogues known as fractional Chern insulators. These topologically ordered phases represent a departure from the conventional Landau symmetry breaking classification of matter, seen in the absence of local order parameters and the presence of global topological properties insensitive to local perturbations. Identifying and manufacturing materials that could host fractional Chern insulator states has a great potential for technological use.

In this thesis, we provide the necessary background required for understanding the results of the accompanying papers [Phys. Rev. Lett. 124, 106803 & Phys. Rev. Lett. 126, 026801]. The theory of fractional Chern insulators is discussed followed by an introduction to the Moiré models used. In the two accompanying papers, we theoretically study a number of flat band Moiré materials aiming at identifying the possible phases that occur at fractional band fillings using a combination of analytical and numerical techniques. By reformulating the problem in terms of holes instead of electrons, it's possible to identify a variety of emergent weakly interacting Fermi liquids from an initial strongly interacting problem. In addition, our findings also include several high temperature fractional Chern insulator states at different fillings without external magnetic field.

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.