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Why nonlocal equations?
All partial differential equations are a limit case of nonlocal equations. A good understanding of nonlocal equations can ultimately provide a better understanding of their limit case: the PDEs. However, there are some cases in which a nonlocal equation gives a significantly better model than a PDE. Some of the most clear examples in which it is necessary to resort to nonlocal equations are

Existence and uniqueness results
For a variety of nonlinear elliptic and parabolic equations, the existence of viscosity solutions can be obtained using Perron's method. The uniqueness of solutions is a consequence of the comparison principle.
There are some equations for which this general framework does not work, for example the surface quasigeostrophic equation. One could say that the underlying reason is that the equation is not purely parabolic, but it has one hyperbolic term.
Regularity results
The regularity tools used for nonlocal equations vary depending on the type of equation.
Nonlinear equations
The starting point to study the regularity of solutions to a nonlinear elliptic or parabolic equation are the Holder estimates which hold under very weak assumptions and rough coefficients. They are related to the Harnack inequality.
For some fully nonlinear integrodifferential equation with continuous coefficients, we can prove $C^{1,\alpha}$ estimates.
Under certain hypothesis, the nonlocal Bellman equation from optimal stochastic control has classical solutions due to the nonlocal version of EvansKrylov theorem.
Semilinear equations
There are several interesting models that are semilinear equations. Those equations consists of either the fractional Laplacian or fractional heat equation plus a nonlinear term.
There are challenging regularity questions especially when the Laplacian interacts with gradient terms in Driftdiffusion equations. A simple method that has been successful in proving the well posedness of some semilinear equations with drift terms in the critical case (when both terms have the same scaling properties) is the conserved modulus of continuity approach, often called "nonlocal maximum principle method".