Here is a list of CFTs. They are organized by dimension. Note that some CFTs, like the gaussian theories and percolation, can be defined in any number of dimensions.

Some special classes of CFTs:

• Holographic CFTs
• Superconformal field theories
• Logarithmic CFTs
• Topological field theories
• Non-unitary CFTs

In 1d, the conformal Ward identity $T_\mu{}^\mu = 0$ implies $T_{00} = H = 0$. Therefore quantum systems which truly respect conformal invariance are non-local (see here for another argument). Nonetheless, there are a number of interesting models exhibiting conformal symmetry or near conformal symmetry in one dimension.

2d CFTs are the best understood class of CFTs, due to their larger Virasoro Algebra

Minimal Models

• 2d Ising
• All the others

Liouville theory

Narain theories

Percolation

There are no known unitary interacting CFTs in $d>6$.

There are constructions of non-unitary theories in higher dimensions (see non-unitary CFTs).

There are no interacting superconformal field theories in $d>6$ because there are no superconformal algebras (satisfying certain assumptions). This follows from the classification of superconformal algebras due to Nahm. See also the proof of Minwalla for a more recent discussion.