  We examine two-dimensional conformal field theories (CFTs) at central charge
c=0. These arise typically in the description of critical systems with quenched
disorder, but also in other contexts including dilute self-avoiding polymers
and percolation. We show that such CFTs must in general possess, in addition to
their stress energy tensor T(z), an extra field whose holomorphic part, t(z),
has conformal weight two. The singular part of the Operator Product Expansion
(OPE) between T(z) and t(z) is uniquely fixed up to a single number b, defining
a new `anomaly' which is a characteristic of any c=0 CFT, and which may be used
to distinguish between different such CFTs. The extra field t(z) is not primary
(unless b=0), and is a so-called `logarithmic operator' except in special cases
which include affine (Kac-Moody) Lie-super current algebras. The number b
controls the question of whether Virasoro null-vectors arising at certain
conformal weights contained in the c=0 Kac table may be set to zero or not, in
these nonunitary theories. This has, in the familiar manner, implications on
the existence of differential equations satisfied by conformal blocks involving
primary operators with Kac-table dimensions. It is shown that c=0 theories
where t(z) is logarithmic, contain, besides T and t, additional fields with
conformal weight two. If the latter are a fermionic pair, the OPEs between the
holomorphic parts of all these conformal weight-two operators are automatically
covariant under a global U(1|1) supersymmetry. A full extension of the Virasoro
algebra by the Laurent modes of these extra conformal weight-two fields,
including t(z), remains an interesting question for future work.
