Spectral properties of CFT

What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac equations, Klein-Gordon equations or even the Maxwell equations...But, how do we work with conformal invariance, either in 2d, where the conformal group is infinite-dimensional or in d>2 where the conformal group is "finite-dimensional"? Note: I think It is related to correlators of conformal fields but I need a more precise and intuitive(if possible) physical/mathematical idea of how we face CFT.

-

If you add the hypothesis of unitarity you have that the conformal weights of your field $\Delta_i$ must obey the following inequality: $$\Delta_i \geq \frac{d-2}{2} + l$$ where $l$ is the spin of the operator.
The conformal bootstrap equation give you also constraints on the coefficients $C_{ijk}$ appearing in the 3-point correlation function.
As far as I know, a CFT is defined by a set $\{\Delta_i,C_{ijk}\}$ and these are the constraints on them.