Abstract: Recently V.I.Arnold introduced three new invariants of a generic immersion of the circle to the plane. These invariants are similar to Vassiliev invariants of classical knots. In a sense they are of degree one. In this paper an investigation based on similar ideas is done for real algebraic plane projective curves. In this more algebraic setting Arnold's invariants have natural counter-parts, two of which admit definitions in terms of the complexification of a curve. On the other hand, the Rokhlin complex orientation formula for a real algebraic curve bounding in its complexification suggests new combinatorial formulas for these two Arnold's invariants. Using the formulas I prove Arnold's conjecture. Arnold's invariants are generalized to generic collections of immersions of the circle to the projective plane and other surfaces. Some invariants of high degrees admitting similar formulas are discussed.