  We analyse global anomalies and related constraints in the Standard Model
(SM) and various Beyond the Standard Model (BSM) theories. We begin by
considering four distinct, but equally valid, versions of the SM, in which the
gauge group is taken to be $G=G_{\text{SM}}/\Gamma_n$, with
$G_{\text{SM}}=SU(3)\times SU(2) \times U(1)$ and $\Gamma_n$ isomorphic to
$\mathbb{Z}/n$ where $n\in\left\{1,2,3,6\right\}$. In addition to deriving
constraints on the hypercharges of fields transforming in arbitrary
representations of the $SU(3)\times SU(2)$ factor, we study the possibility of
global anomalies in theories with these gauge groups by computing the bordism
groups $\Omega^{\text{Spin}}_5(BG)$ using the Atiyah-Hirzebruch spectral
sequence. In two cases we show that there are no global anomalies beyond the
Witten anomaly, while in the other cases we show that there are no global
anomalies at all, illustrating the subtle interplay between local and global
anomalies. While freedom from global anomalies has been previously shown for
the specific fermion content of the SM by embedding the SM in an anomaly-free
$SU(5)$ GUT, our results here remain true when the SM fermion content is
extended arbitrarily. Going beyond the SM gauge groups, we show that there are
no new global anomalies in extensions of the (usual) SM gauge group by $U(1)^m$
for any integer $m$, which correspond to phenomenologically well-motivated BSM
theories featuring multiple $Z^\prime$ bosons. Nor do we find any new global
anomalies in various grand unified theories, including Pati-Salam and
trinification models. We also consider global anomalies in a family of theories
with gauge group $SU(N)\times Sp(M)\times U(1)$, which share the phase
structure of the SM for certain $(N, M)$.
