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
• Nonrelativistic 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. 2d CFTs differ based on their central charge; those with $c<1$ comprise considerably simpler class of Minimal Models.

Those with $c>1$ are more complicated and in general are not fully classified yet.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.31.851

https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.122.211601

https://arxiv.org/abs/hep-th/0608014

https://arxiv.org/abs/1805.06467

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.

Non-chiral CFTs can often be defined in continuous dimension $d$, though they are usually explicilty non-unitary for noninteger $d$ due to evanescent operators that have negative norm 1. In the limit as $d$ approaches an integer, these evanescent operators decouple from the CFT.

These CFTs apparently exist, even nonperturbatively. What's the conformal symmetry group? Possibly relevant: http://mr.crossref.org/iPage?doi=10.1070%2FRM1988v043n02ABEH001720

Here's an example of the Ising model on the Sierpinski carpet: https://arxiv.org/abs/cond-mat/9802018.

Here's some bootstrap work: https://arxiv.org/abs/1309.5089 2 3 4.

There is a possible connection to non-conformal D$p$-brane holography http://arxiv.org/abs/2503.18770.

It is possible to define quantum field theories with full SO$(d+1,1)$ symmetry that do not have a local action (i.e. $S= \int d^d x O(x)$ for some local operator $x$). These are called nonlocal CFTs, long-range CFTs, or just Conformal Theories (CTs).

The simplest example of these are the generalized free fields (GFFs, also called mean field theory), which are Gaussian: we declare one field to have a conformal two-point function, and then determine all other correlators by the usual Wick contractions.

By perturbing one of these GFFs $\phi$ of arbitrary scaling dimension $\Delta$ by $\phi^4$ and tuning to the critical point, we can construct the the long-range Ising CFT in arbitrary dimension $d$. The long-range Ising CFT becomes the standard Ising CFT (+ a decoupled free field) in the limit $\Delta \to \Delta_{\phi, \text{Ising}}$ (i.e. $1/8$ in $d=2$, $0.518$ in $d=3$) 1.

From the perspective of conformal data, the signal that these theories are nonlocal is that they do not possess a local stress tensor. That is, in the OPE of these theories there is no conserved spin-2 operator. This is reasonable because the stress tensor is an operator measuring the response of the theory to a change of the geometry. If the theory is nonlocal, then the response will not be local, so $T^{\mu\nu}$ cannot be a local operator.