  It is known that a subset of fractional quantum Hall wave functions has been
expressed as conformal field theory (CFT) correlators, notably the Laughlin
wave function at filling factor $\nu=1/m$ ($m$ odd) and its quasiholes, and the
Pfaffian wave function at $\nu=1/2$ and its quasiholes. We develop a general
scheme for constructing composite-fermion (CF) wave functions from conformal
field theory. Quasiparticles at $\nu=1/m$ are created by inserting anyonic
vertex operators, $P_{\frac{1}{m}}(z)$, that replace a subset of the electron
operators in the correlator. The one-quasiparticle wave function is identical
to the corresponding CF wave function, and the two-quasiparticle wave function
has correct fractional charge and statistics and is numerically almost
identical to the corresponding CF wave function. We further show how to exactly
represent the CF wavefunctions in the Jain series $\nu = s/(2sp+1)$ as the CFT
correlators of a new type of fermionic vertex operators, $V_{p,n}(z)$,
constructed from $n$ free compactified bosons; these operators provide the CFT
representation of composite fermions carrying $2p$ flux quanta in the $n^{\rm
th}$ CF Landau level. We also construct the corresponding quasiparticle- and
quasihole operators and argue that they have the expected fractional charge and
statistics. For filling fractions 2/5 and 3/7 we show that the chiral CFTs that
describe the bulk wave functions are identical to those given by Wen's general
classification of quantum Hall states in terms of $K$-matrices and $l$- and
$t$-vectors, and we propose that to be generally true. Our results suggest a
general procedure for constructing quasiparticle wave functions for other
fractional Hall states, as well as for constructing ground states at filling
fractions not contained in the principal Jain series.
