The behaviour of fluids is beautiful, complex and difficult to predict. If a viscous liquid containing gas bubbles is driven through a confined geometry, it exhibits complex nonlinear behaviour, even in the absence of fluid inertia. The behaviour is driven by interaction between the geometry, capillary forces at the liquid-gas interface, and viscous forces within the bulk liquid. The system is relatively simple, yet it exhibits a remarkable variety of nonlinear phenomena, making it ideally suited to studies of bifurcations and instabilities in fluid dynamics. At the Manchester Centre for Nonlinear Dynamics, we use a combined experimental and theoretical approach to study such flows through Hele-Shaw channels with small axially-uniform variations in channel depth. From a theoretical point of view, the system is attractive because it can be accurately described by depth-averaged equations. Moreover, the state of the system is characterised by the bubbles' shapes and relative locations within the channel, which are easy to access experimentally. An unusual feature of the system is its frequent topology changes as bubbles break up into smaller bubbles that can later recombine or not. Topology changes alter the entire solution structure, meaning that a single bifurcation diagram cannot capture the system's dynamics. For a fixed set of control parameters (e.g. flow rate and total bubble volume) the solution structure can change several times over the course of an experiment. In addition, the system shows heightened sensitivity to perturbations in regimes where the bubble is likely to break up. We find multiple steadily propagating and oscillatory states, regions of multi-stability and practical unpredictability, where the final outcome is extremely sensitive to minor perturbations. We use numerical bifurcation techniques to determine the structures that organise the dynamics of the system, including key roles played by unstable periodic orbits and weakly unstable steadily propagating solutions. We find a strong correspondence between single and multiple bubble solutions which allows us to predict and construct non-trivial stable multi-bubble states. Although our interest in these systems is motivated by the desire to understand complex nonlinear dynamics, there are applications to microfluidics. In this talk, I will present a survey of what we know about the system so far, it's connection to other interfacial flows and present some of the open challenges.