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It is well known that the complex analytical properties of a real submanifold M in the complex space C n are most accessible through consideration of the complex tangents to M. The properties we have in mind are related to the behavior of holomorphic functions on or near M and to the behavior of M under biholomorphic transformation. The case in which M is a real hypersurface is most familiar, while much less is known for higher codimension. In this paper we consider the critical case of a real ndimensional manifold M in C n, which we also assume to be real analytic. At a generic point M is locally equivalent to the standard R n in C n. However, near a complex tangent M may aquire a non-trivial local hull of holomorphy and other biholomorphic invariants. We begin with the simplest non-trivial case, which is a surface M2cC 2 with an isolated, suitably non-degenerate complex tangent. Here one already encounters a rich structure and non-trivial problems. In coordinates zj=xi+iy i, j= 1,2, M may be written locally as

Normal forms for real surfaces in C2 near complex tangents and hyperbolic surface transformations