  Luttinger's theorem connects a basic microscopic property of a given metallic
crystalline material, the number of electrons per unit cell, to the volume,
enclosed by its Fermi surface, which defines its low-energy observable
properties. Such statements are valuable since, in general, deducing a
low-energy description from microscopics, which may perhaps be regarded as the
main problem of condensed matter theory, is far from easy. In this paper we
present a unified framework, which allows one to discuss Luttinger theorems for
ordinary metals, as well as closely analogous exact statements for topological
(semi)metals, whose low-energy description contains either discrete point or
continuous line nodes. This framework is based on the 't Hooft anomaly of the
emergent charge conservation symmetry at each point on the Fermi surface, a
concept recently proposed by Else, Thorngren and Senthil [Phys. Rev. X 11,
021005 (2021)]. We find that the Fermi surface codimension $p$ plays a crucial
role for the emergent anomaly. For odd $p$, such as ordinary metals ($p=1$) and
magnetic Weyl semimetals ($p=3$), the emergent symmetry has a generalized
chiral anomaly. For even $p$, such as graphene and nodal line semimetals (both
with $p=2$), the emergent symmetry has a generalized parity anomaly. When
restricted to microscopic symmetries, such as $U(1)$ and lattice symmetries,
the emergent anomalies imply (generalized) Luttinger theorems, relating Fermi
surface volume to various topological responses. The corresponding topological
responses are the charge density for $p=1$, Hall conductivity for $p=3$, and
polarization for $p=2$. As a by-product of our results, we clarify exactly what
is anomalous about the surface states of nodal line semimetals.
