  Traversable wormholes necessarily require violations of the averaged null
energy condition; this being the definition of ``exotic matter''. However, the
theorems which guarantee the energy condition violation are remarkably silent
when it comes to making quantitative statements regarding the ``total amount''
of energy condition violating matter in the spacetime. We develop a suitable
measure for quantifying this notion, and demonstrate the existence of spacetime
geometries containing traversable wormholes that are supported by arbitrarily
small quantities of ``exotic matter''.
