  While chiral quantum field theories (QFTs) describe a wide range of physical
systems, from the standard model to topological quantum matter, the realization
of chiral QFTs on a lattice has proved to be difficult due to the
Nielsen-Ninomiya theorem and the possible presence of quantum anomalies. In
this thesis, we use the connection between chiral phases of matter and chiral
quantum field theories (QFTs) to define chiral QFTs on a lattice and allow a
huge class of exotic field theories to be simulated numerically. Our work
builds on the 'mirror fermion' approach to the problem of defining chiral
theories on a lattice, which defines chiral field theories as the edge modes of
chiral phases. We begin by reviewing the deep connections between chiral phases
of matter, chiral field theories, and anomalies. We then develop numerical
treatments of an $SU(2)$ chiral field theory, and provide a semiclassically
solvable definition of Abelian $2+1$ chiral topological orders. This leads to
an exactly solvable definition of chiral $U(1)$ SPT phases with zero
correlation length, which we use to extract the edge chiral field theories
exactly. These zero-correlation length models are vastly more simple than
previous approaches to defining chiral field theories on the lattice.
