The main aim of these notes is to record some examples of conformal field theories in terms which are familiar to `old-fashioned' quantum field theorists. Basically what we show here is that many of the `examples' which have appeared in the literature can be easily understood in terms of representations of the algebras of the canonical commutation (CCR) and anticommutation relations (CAR). The representations we describe here are of the sort called quasi-free that is, there is no interaction. Their interest lies in their connection with other mathematical topics (Riemann surfaces and theta functions in particular). Indeed as we shall see the \(n\)-point (or correlation) functions of these representations possess holomorphy properties far more interesting than those arising from representations of the CCR or CAR so far considered. Our motivation lies not in the direction of interacting field theories in two dimensions as such but in the conformal invariance of two dimensional models in statistical mechanics. It has been our view that an understanding of these matters may be facilitated by considering field theories on the spectral curves associated with various transfer matrices. In order for this view to make sense it is essential that we provide some meaning for the therm `field theory on a spectral curve'. We will, on the basis of our examples below, give a general discussion of what we mean by this in the final section. As far as the existing literature is concerned (with the exception of the general framework proposed by Segal) it seems that little attention has been paid to the usual questions an old-fashioned quantum field theorist might ask. Principally we are thinking of the questions: what is the Hilbert space of states, how do the fields act and do the \(n\)-point functions which have been explicitly given satisfy the usual positivity requirements of the field theory? This work is an extension of previous collaborations with Mike Eastwood and John Palmer whom we thank for many discussions.