  In the tangent plane at any point of a surface in the four-dimensional
Euclidean space we consider an invariant linear map of Weingarten-type and find
a geometrically determined moving frame field. Writing derivative formulas of
Frenet-type for this frame field, we obtain eight invariant functions. We prove
a fundamental theorem of Bonnet-type, stating that these eight invariants under
some natural conditions determine the surface up to a motion.
  We show that the basic geometric classes of surfaces in the four-dimensional
Euclidean space, determined by conditions on their invariants, can be
interpreted in terms of the properties of two geometric figures: the tangent
indicatrix, which is a conic in the tangent plane, and the normal curvature
ellipse.
  We construct a family of surfaces with flat normal connection.
