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The microtubule cytoskeleton is a self-organising system of microtubule filaments, molecular motors and regulatory proteins that plays a number of vital roles in all eukaryotic cells, acting as a scaffold to provide structural support to the cell and forming tracks for intracellular transport. Arguably one of its most important roles is in assembling the mitotic spindle, the highly dynamic force-generating structure that segregates chromosomes during mitosis. Critical to the spindle's dynamism is the dynamic instability of microtubules. At the heart of the long-standing GTP cap model of dynamic instability is the notion that stochastic fluctuations of the GTP cap drive microtubules to catastrophe, but a quantitative understanding of GTP cap fluctuations is missing. In Chapter 2 a simple mathematical model is developed that predicts microtubule fluctuation behaviour in the steady-state. The theory is validated by data, which suggests that microtubules are remarkably stable during most of their lifetime. How microtubules and motors self-organise into the complex, bipolar architecture of the spindle is not well understood. In Chapter 3 a microscopic computational model of microtubule-motor self-organisation is developed based on in vitro experiments. Two control parameters are identified capturing system composition and kinetics that predict the outcome of self-organisation and uncover generic, mechanistic rules governing active filament-motor networks. Design principles of spindle bipolarity are elucidated which highlight the combined effect of motor activity and filament dynamics on self-organisation. Computer simulations are an increasingly powerful tool for modelling the cytoskeleton, but existing models differ in key respects. In Chapter 4 the effects of spatial dimensionality and the inclusion/omission of steric interactions on simulated microtubule-motor networks is systematically evaluated. Spatial dimensionality and steric interactions are found to have a combined effect on the probability of microtubule crossings, which in turn play critical and distinct roles in several archetypal microtubule-motor networks. Guidelines for appropriate modelling choices are discussed.