  In a series of papers, we have shown that from the representatio theory of a
compact groupoid one can reconstruct the groupoid using the procedure similar
to the Tannaka-Krein duality for compact groups. In this part we study
continuous representations of compact groupoids. We show that irreducible
representations have finite dimensional fibres. We prove the Schur's lemma,
Gelfand-Raikov theorem and Peter-Weyl theorem for compact groupoids.
