Abstract - Let X be a holomorphic symplectic manifold and D a smooth hypersurface in X. Then the restriction of the symplectic form on D has one-dimensional kernel at each point. This distribution is called the characteristic foliation. I shall survey a few results concerning the possible Zariski closure of a general leaf of this foliation by myself and Campana, myself and my former student L. Guseva, and more recently by my student R. Abugaliev.