The first main result of this paper is a unified approach on the studies of Orlicz-Morrey spaces. There are at least three versions of Orlicz-Morrey spaces, the Nakai type, the Sawano-Sugano-Tanaka type and the Guliyev-Hasanov-Sawano-Noi type. This paper unifies the studies by introducing a Orlicz-Morrey spaces that include those Orlicz-Morrey spaces mentioned above. The second main result is on the applications of the Orlicz-Morrey spaces on partial differential equations. We obtain the Agmon-Douglis-Nirenberg estimates of uniformly elliptic equations, the estimates of the solutions of some nonhomogeneous quasilinear elliptic equations and the Beltrami equations on Orlicz-Morrey spaces. Copyright © 2022 The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.