  We propose a broad class of $d$-dimensional conformal field theories of
$SU(N)$ adjoint scalar fields generalising the 4$d$ Fishnet CFT (FCFT)
discovered by \"O. G\"urdogan and one of the authors as a special limit of
$\gamma$-deformed $\mathcal{N}=4$ SYM theory. In the planar $N\to\infty $ limit
the FCFTs are dominated by the ``fishnet" planar Feynman graphs. These graphs
are explicitly integrable, as was shown long ago by A. Zamolodchikov. The
Zamolodchikov's construction, based on the dual Baxter lattice (straight lines
on the plane intersecting at arbitrary slopes) and the star-triangle
identities, can serve as a ``loom" for ``weaving" the Feynman graphs of these
FCFTs, with certain types of propagators, at any $d$. The Baxter lattice with
$M$ different slopes and any number of lines parallel to those, generates an
FCFT consisting of $M(M-1)$ fields and a certain number of chiral vertices of
different valences with distinguished couplings. These non-unitary, logarithmic
CFTs enjoy certain reality properties for their spectrum due to a symmetry
similar to the PT-invariance of non-hermitian hamiltonians proposed by C.
Bender. We discuss in more detail the theories generated by a loom with
$M=2,3,4$, and the generalisation of the loom FCFTs for spinning fields in
4$d$.
