The paper characterizes Miura surfaces defined as smooth surfaces that the regular periodic Miura ori can fit in the limit where the size of the creases is infinitely small compared to the typical radius of curvature. Based on a model due to Schenk and Guest, the Miura crease pattern is enriched so as to allow access to non-planar configurations. In contrast with previous work where the pattern is modified in order to fit different target surfaces, here we are interested in the converse problem of determining all the surfaces that can be fitted by one and the same pattern. The central result is a constrained nonlinear partial differential equation satisfied by the parametrization of any Miura surface. As an application, examples of bounded and unbounded Miura surfaces are presented along with a complete classification of axisymmetric and ruled ones.