  We introduce unitary representations of continuous groupoids on continuous
fields of Hilbert spaces. We investigate some properties of these objects and
discuss some of the standard constructions from representation theory in this
particular context. An important r\^{ole} is played by the regular
representation. We conclude by discussing some operator algebra associated to
continuous representations of groupoids; in particular, we analyse the
relationship of continuous representations of $G$ and continuous
representations of the Banach $*$-category $\hat{L}^{1}(G)$.
