  We discuss five-dimensional supersymmetric gauge theories. An anomaly renders
some theories inconsistent and others consistent only upon including a
Wess-Zumino type Chern-Simons term. We discuss some necessary conditions for
existence of nontrivial renormalization group fixed points and find all
possible gauge groups and matter content which satisfy them. In some cases, the
existence of these fixed points can be inferred from string duality
considerations. In other cases, they arise from M-theory on Calabi-Yau
threefolds. We explore connections between aspects of the gauge theory and
Calabi-Yau geometry. A consequence of our classification of field theories with
nontrivial fixed points is a fairly complete classification of a class of
singularities of Calabi-Yau threefolds which generalize the ``del Pezzo
contractions'' and occur at higher codimension walls of the K\"{a}hler cone.
