We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic 0, prove almost all cases in positive characteristic, and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation theory of curves on K3 surfaces. The first, regeneration, is a process opposite to specialization, which preserves the geometric genus and does not require the class of the curve to extend. The second, called the marked point trick, allows for a controlled degeneration of rational curves to integral ones in certain situations. Combining the two proves existence of integral curves of unbounded degree of any geometric genus g for any projective K3 surface in characteristic 0.