In the first part of this disseration, I present my work on the emergence of Kac– Moody symmetry in critical quantum spin chains. Given a critical quantum spin chain with a microscopic Lie-group symmetry, corresponding e.g. to U(1) or S U (2) spin isotropy, the emergence of Kac–Moody symmetry is numerically investigated at low energies and long distances. In that regime, one such critical quantum spin chain is described by a conformal field theory where the usual Virasoro algebra associated to conformal invariance is augmented with a Kac– Moody algebra associated to conserved currents. Specifically, we first propose a method to construct lattice operators corresponding to the Kac–Moody gen- erators. It is then numerically verified that, when projected onto low energy states of the quantum spin chain, these operators indeed approximately fulfill the Kac–Moody algebra. The lattice version of the Kac–Moody generators al- low us to compute the so-called level constant and to organize the low-energy eigenstates of the lattice Hamiltonian into Kac–Moody towers. This proposal is illustrated with the XXZ model and the Heisenberg model with a next-to- nearest-neighbor coupling. In the second part of this dissertation, I discuss my study of a general im- plementation of the Virasoro generators and Kac–Moody currents in generic tensor network representations of 2-dimensional critical lattice models. This proposal works even when a quantum Hamiltonian of the lattice model is not available, which is the case in many numerical computations involving numer- ical blockings. This proposal is tested on the 2d Ising model, and also the dimer model, which works to high accuracy even with a fairly small system size. As the method proposed here only makes use of eigenstates of a small cylinder to generate descendant states in a larger cylinder, it suggests some intricate alge- braic relations between lattice of different sizes.