This thesis is devoted to the study of topological effects in quantum field theories, with a particular focus on phenomenological applications. We begin by deriving a general classification of topological terms appearing in a non-linear sigma model based on maps from an arbitrary worldvolume manifold to a homogeneous space $G/H$ (where $G$ is an arbitrary Lie group and $H\subset G$). Such models are ubiquitous in phenomenology; in three or more dimensions they cover all cases in which only some subgroup $H$ of a dynamical symmetry group $G$ is linearly realized in vacuo. The classification is based on the observation that, for topological terms, the maps from the worldvolume to $G/H$ may be replaced by singular homology cycles on $G/H$. We find that such terms come in one of two types, which we refer to as Aharonov-Bohm' (AB) and Wess-Zumino' (WZ) terms. We derive a new condition for their $G$-invariance, which we call the `Manton condition', which is necessary and sufficient when the Lie group $G$ is connected.

Armed with this classification of topological terms, we then apply it to Composite Higgs models based on a variety of coset spaces $G/H$ and discuss their phenomenology. For example, we point out the existence of an AB term in the minimal Composite Higgs model based on $SO(5)/SO(4)$, whose phenomenological effects arise only at the non-perturbative level, and lead to $P$ and $CP$ violation in the Higgs sector. Consideration of the Manton condition leads us to discover a rather subtle anomaly in a non-minimal model based on $SO(5)\times U(1)/SO(4)$ (a model which does, however, feature an AB term not previously noticed in the literature). A particularly rich topological structure, with six distinct terms of various types, is uncovered for the model based on $SO(6)/SO(4)$, which features two Higgs doublets and one singlet. Perhaps most importantly for phenomenology, measuring the coefficients of WZ terms that appear in any of these Composite Higgs models can allow one to probe the gauge group of the underlying microscopic theory.

As a further application of our results, we analyse quantum mechanics models featuring such topological terms. In this context, a topological term couples the particle to a background magnetic field. The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. We show how both locality and symmetry can be made manifest, by passing to an otherwise redundant description on a principal bundle over the original configuration space, and by promoting the original symmetry group to a central extension thereof. We then demonstrate how harmonic analysis on the extended symmetry group can be used to solve the Schr{"o}dinger equation.

To conclude our study of topological terms in sigma models, we show that the classification we have proposed may be rigorously justified (and generalised) using differential cohomology theory. In doing so, we introduce the notion of the `$G$-invariant differential characters' of a manifold $M$. Within this language, the Manton condition follows from the homotopy formula for differential characters, and we show that it remains necessary and sufficient under weaker conditions than connectedness of $G$. We prove that the abelian group of $G$-invariant differential characters sits inside various exact sequences and commutative diagrams, which thus provide us with some powerful algebraic tools for classifying topological terms in quantum field theories.

In the remainder of the thesis we depart from the topic of sigma models and turn to gauge theories. We analyse anomalies (which may be understood as arising from topological effects) in both the Standard Model (SM) and theories Beyond the Standard Model (BSM). This analysis consists of two parts, in which we consider local' and global' anomalies in a gauge symmetry $G$; the former depend only on the Lie algebra of $G$, while the latter are sensitive also to its global structure, {\em i.e.} its topology.

We first chart the space of anomaly-free extensions of the SM by a flavour-dependent $U(1)$ gauge symmetry, using arithmetic techniques from Diophantine analysis to cancel all possible local anomalies. We then develop some of these anomaly-free theories into phenomenological models featuring a heavy $Z\prime$ gauge boson, that can account for a collection of recent measurements involving $b\to s\mu \mu$ transitions which are discrepant with SM predictions. We discuss how these models might also explain coarse features of the fermion mass problem, such as the heaviness of the third family.

We then turn to global anomalies, which we analyse using the Dai-Freed theorem. Our principal tool here is to compute the bordism groups of the classifying spaces of various Lie groups, preserving particular spin structures, using the Atiyah-Hirzebruch spectral sequence. We show that there are no global anomalies (beyond the Witten anomaly associated with the electroweak factor) in four different `versions' of the SM, in which the gauge group is taken to be $GSM/\Gamma$, with $GSM=SU(3)\times SU(2)\times U(1)$ and $\Gamma \in \{0,Z2,Z3,Z6\}$. We also show that there are no new global anomalies in $U(1)m$ extensions of the SM, which feature multiple $Z\prime$ bosons, or in the Pati-Salam model.