It is well-known that the classification of flat surfaces in Euclidean 3-space is one of the most basic results in differential geometry. For surfaces in the complex Euclidean plane \bm{C}
2 endowed with almost complex structure J, flat surfaces are the simplest ones from intrinsic point of views. On the other hand, from J-action point of views, the most natural surfaces in \bm{C}
2 are slant surfaces, i.e., surfaces with constant Wintinger angle. In this paper the author completely classifies flat slant surfaces in \bm{C}
2. The main result states that, beside the totally geodesic ones, there are five large classes of flat slant surfaces in \bm{C}
2. Conversely, every non-totally geodesic flat slant surfaces in \bm{C}
2 is locally a surface given by these five classes.
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