An immersion $f: \Sigma \to \mathbb{CP}^2$ is called a minimal Lagrangian surface if it is minimal with respect to the Fubini study metric and Lagrangian with respect to the Kähler form. Besides the real projective plane and minimal Lagrangian tori, which can all be constructed via integrable systems methods, the only known compact examples have been obtained by Haskins and Kapouleas for odd genera. In this talk, we explain the construction of new compact minimal Lagrangian surfaces of genus $g=\tfrac{(k-2)(k-1)}{2}$ for large $k \in \mathbb{N}$ using gauge theoretic and loop group factorization methods. These surfaces are analogous to Lawson’s minimal surfaces in the 3-sphere and coincide with the projective plane and the Clifford torus for $k=2, 3${}, respectively. We determine their symmetry groups and show that the underlying Riemann surfaces are the Fermat curves. We also discuss further geometric properties such as their area and Willmore energy. This talked is based on joint work with Charles Ouyang and Franz Pedit.