  We prove that the Gauss curvature and the curvature of the normal connection
of any minimal surface in the four dimensional Euclidean space satisfy an
inequality, which generates two classes of minimal surfaces: minimal surfaces
of general type and minimal super-conformal surfaces. We prove a Bonnet-type
theorem for strongly regular minimal surfaces of general type in terms of their
invariants. We introduce canonical parameters on strongly regular minimal
surfaces of general type and prove that any such a surface is determined up to
a motion by two invariant functions satisfying a system of two natural partial
differential equations. On any minimal surface of the basic class of non
strongly regular minimal surfaces we define canonical parameters and prove that
any such a surface is determined up to a motion by two invariant functions of
one variable satisfying a system of two natural ordinary differential
equations. We find a geometric description of this class of non strongly
regular minimal surfaces.
