  We present a comprehensive study of the effective Conformal Field Theory
(CFT) describing the low energy excitations of a gas of spinless interacting
fermions on a circle in the gapless regime (Luttinger liquid). Functional
techniques and modular transformation properties are used to compute all
correlation functions in a finite size and at finite temperature. Forward
scattering disorder is treated exactly. Laughlin experiments on charge
transport in a Quantum Hall Fluid on a cylinder are reviewed within this CFT
framework. Edge excitations above a given bulk excitation are described by a
twisted version of the Luttinger effective theory. Luttinger CFTs corresponding
to the nu =1/(2p+1) filling fractions appear to be rational CFTs (RCFT).
Generators of the extended symmetry algebra are identified as edge fermions
creators and annihilators, thus giving a physical meaning to the RCFT point of
view on edge excitations of these sequences.
