In order to study their temporal dynamics, the size, density or abundance of populations are often monitored over discrete time steps. Many of these populations tend to have internal structure or interconnections that affect individuals at various spatiotemporal scales, such as developmental stages, resource preferences, group interactions and movement. In this thesiswe are mainly concerned with modelling the dynamics of such structured populations in discrete-time, from both deterministic and stochastic perspectives. In Chapter 1 we motivate the problems we are interested in throughout this thesis. In Chapter 2 we give some technical background needed in order for this thesis to be reasonably self-contained. In Chapter 3we discuss various existing frameworks and results within the literature related to structured population models, while also demonstrating how matrix stability plays an important role in understanding such systems. In Chapter 4 we propose a costless, densitydependent invasion model, where a population moves between two resources. We explore some of the properties of this model, both theoretically and numerically, in the context of a species expanding its habitat or range. We then apply this model to an pest case study in order to further understand how host switching can affect long-term population viability. In Chapter 5 we propose a model of costly, density-dependent dispersal between a finite number of regions. We study its stability and persistence properties, and numerically show howit relates to various source-sink scenarios. In Chapter 6we focus on deriving sufficient conditions for the stability of the extinction equilibrium for coupled linear time-invariant systems, which is robust under diffusive couplings, so-called robust diffusive stability (RDS). This model corresponds to demographically-structured populations diffusively dispersing between habitats, such as species migration for example. We discuss the role of the existence/nonexistence of copositive, quadratic and diagonal Lyapunov functions in determining RDS. We then discuss the anithesis of RDS, diffusive growth. Throughout, we apply our results to some commonly used matrix population models. In Chapter 7 we propose a stochastic model that aims to capture the interactions between animal groups within a social population. We conduct simulation scenarios and fit this model to real-world data, to show its applicability. We then discuss the findings of this model fit in the context of ecological theory. We derive an approximation to the marginal group correlation for a simpler model, which describes their net interactions over an observation period. We then theoretically discuss its interpretation in multiple predator-prey scenarios. Finally, in Chapter 8, we conclude by discussing various extensions and open questions suggested by the results presented throughout this thesis.