- Let S be a compact oriented surface. We construct homogeneous quasimorphisms on Diff (S, area), on Diff0 (S, area) and on Ham (S) generalizing the constructions of Gambaudo- Ghys and Polterovich. We prove that there are infinitely many linearly independent homogeneous quasimorphisms on Diff (S, area), on Diff0 (S, area) and on Ham (S) whose absolute values bound from below the topological entropy. In case when S has a positive genus, the quasimorphisms we construct on Ham (S) are C0-continuous. We define a bi- invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on Ham (S) is unbounded.