  We introduce a new class of two dimensional conformal field theories by
extending Wess-Zumino-Witten (WZW) models to chiral algebras with matrix-valued
levels. The new CFTs are based on holomorphic currents with an operator product
expansion characterized by a positive integer-valued matrix $K_{AB}$. We use
the Sugawara construction to compute the energy-momentum tensor, the central
charge, and the spectrum of conformal dimensions of the CFTs based on this
algebra. We also construct a set of genus-$1$ characters and show that they
fulfil a representation of the modular group $\text{SL}(2,\mathbb{Z})$ up to a
modular anomaly.
