Let \(N\) be a Kählerian manifold with complex structure \(J\). A submanifold \(M\subset N\) is said to be Lagrangian, if \(JT_pM\) coincides with the normal space of \(M\) at \(p\) for every point \(p\in M\). The author describes 19 types of constant curvature Lagrangian surfaces of the flat Kählerian manifold \(\mathbb{C}^2\) and shows that any constant curvature Lagrangian surface of \(\mathbb{C}^2\) is almost everywhere locally of one of these types. In two other articles the author has also classified the constant curvature Lagrangian surfaces of the two-dimensional complex projective resp. hyperbolic space, see [J. Geom. Phys. 53, No. 4, 428–460 (2005; Zbl 1072.53027) and ibid. 55, No. 4, 399–439 (2005; Zbl 1080.53076)].