Arnold's seminar October 26, 1999 Yu. Chekanov "Proof of Arnold's 4 cusp conjecture" (joint work with P. Pushkar) We give an outline of the proof of the Arnold's conjecture claiming the following: in every family of planar co-oriented fronts without positive self-tangencies which connects a circle in the plane to a circle with the opposite orientation, there exists a front with (at least) 4 singular points. The key ingredient is the theory of decompositions of fronts in J^1(S^1), which also allows to construct nontrivial invariants of Legendrian knots (links).