In this talk, I will explain the notion of anisotropic gravitational Weyl anomalies in Lifshitz field theories and their potential applications. Anomalies are quantum violations of classical symmetries, in this case under a non-relativistic (non-Lorentz-invariant) symmetry group. More specifically, I will focus on the only anomaly found in 1+1-dimensional Lifshitz field theories that arise after coupling to Newton-Cartan geometry with torsion, and it's relation to the Lorentz (diffeomorphism) anomaly in the quantum effective action of a 1+1 CFT. At the heart of this discussion, I will emphasize the role of temporal torsion in generating this anomaly and show how such an anomaly can be derived from a 2+1-dimensional Weyl-invariant Chern-Simons Horava-Lifshitz theory of gravity. I will finally comment briefly on how the 1+1 Lifshitz Weyl anomaly can potentially be connected to thermal Hall transport and some other potential applications.