  We study five-dimensional supersymmetric field theories with one-dimensional
Coulomb branch. We extend a previous analysis which led to non-trivial fixed
points with $E_n$ symmetry ($E_8$, $E_7$, $E_6$, $E_5=Spin(10)$, $E_4=SU(5)$,
$E_3=SU(3)\times SU(2)$, $E_2=SU(2)\times U(1)$ and $E_1=SU(2)$) by finding two
new theories: $\tilde E_1$ with $U(1)$ symmetry and $E_0$ with no symmetry. The
latter is a non-trivial theory with no relevant operators preserving the
super-Poincar\'e symmetry. In terms of string theory these new field theories
enable us to describe compactifications of the type I' theory on $S^1/Z_2$ with
16, 17 or 18 background D8-branes. These theories also play a crucial role in
compactifications of M-theory on Calabi--Yau spaces, providing physical models
for the contractions of del Pezzo surfaces to points (thereby completing the
classification of singularities which can occur at codimension one in K\"ahler
moduli). The structure of the Higgs branch yields a prediction which unifies
the known mathematical facts about del Pezzo transitions in a quite remarkable
way.
