Abstract: In this talk we consider problems involving fractional-order operators on bounded domains. We first review the linear integral fractional Laplacian problem, discuss the regularity solutions and the convergence rates of the standard conforming finite element method. Next, we introduce a ``DG'' formulation motivated by an integration by parts formula for fractional Laplacian and derive the convergence rates of the nonconforming scheme. We also study two nonlinear problems: the nonlocal minimal graph problem and the fractional p-Laplacian. We discuss regularity of solutions to these problems, propose finite element methods to numerically solve them, and analyze their convergence properties.