This paper is a study of summability methods that are based on the Riemann Zeta function. A limitation theorem is proved which gives a necessary condition for a sequence x to be zeta summable. A zeta summability matrix Z t associated with a real sequence t is introduced; a necessary and sufficient condition on the sequence t such that Z t maps l 1 to l 1 is established. Results comparing the strength of the zeta method to that of well-known summability methods are also investigated.