The paper initiates a systematic study of semisimple weak Hopf algebras. Analogues of several results for semisimple Hopf algebras are obtained. A canonical left integral on a weak Hopf algebra
is defined and used to characterize the semisimplicity of
be a semisimple weak Hopf algebra. The quantum and Frobenius-Perron dimensions are studied for finite dimensional
-modules. The Grothendieck ring of
is studied and the class equation is extended for
. Analogues of the trace formulas of Larson and Radford are proved and used to show that the categorical dimension of
divides its Frobenius-Perron dimension in the ring of algebraic integers. Module algebras and coalgebras over
are studied. The author indicates that this paper is a step in the direction of classifying semisimple weak Hopf algebras. This classification should include the classification of fusion categories and module categories over them.