  Nonrelativistic string theory is a self-contained corner of string theory,
with its string spectrum enjoying a Galilean-invariant dispersion relation.
This theory is unitary and ultraviolet complete, and can be studied from first
principles. In these notes, we focus on the bosonic closed string sector. In
curved spacetime, nonrelativistic string theory is defined by a renormalizable
quantum nonlinear sigma model in background fields, following certain symmetry
principles that disallow any deformation towards relativistic string theory. We
review previous proposals of such symmetry principles and propose a modified
version that might be useful for supersymmetrizations. The appropriate
target-space geometry determined by these local spacetime symmetries is string
Newton-Cartan geometry. This geometry is equipped with a two-dimensional
foliation structure that is restricted by torsional constraints. Breaking the
symmetries that give rise to such torsional constraints in the target space
will in general generate quantum corrections to a marginal deformation in the
worldsheet quantum field theory. Such a deformation induces a renormalization
group flow towards sigma models that describe relativistic strings.
