This journal format thesis presents interdisciplinary research on characterising the biological process of intracellular transport using random walk models. The first two chapters outline the aims of the thesis, summarise the main chapters, and introduce intracellular transport, anomalous diffusion and various random walk models. This includes uncoupled continuous time random walks both in discrete and continuous space with extension to Levy walks, the fractional diffusion equation using the structured density approach, velocity random walks and fractional Brownian motion. The third chapter outlines experimental observations for power-law distributed running times in active intracellular transport. In addition, a mesoscopic Levy walk model is introduced making connections to a previously established microscopic framework involving multiple detachment and attachment rates of motor proteins. This work shows that the Levy walk model is capable of accurately describing intracellular transport and is compatible with the microscopic mechanisms of previously proposed theories. Furthermore, a new persistent random walk model with finite velocities that describe the phenomena of self-reinforcing directionality in the transport of endosomes is presented. Through this model, superdiffusion is exhibited despite exponential running times. Exact analytical expressions for the first and second moments are derived and validated using Monte Carlo simulations. The fourth chapter models the passive intracellular transport of lysosomes using the variable-order fractional diffusion equation. In doing so, the asymptotic representation of the solution of the variable-order fractional diffusion equation is found for an anomalous exponent that monotonically increases with position away from the origin. Furthermore, the asymptotic representation is validated using Monte Carlo simulations and experimental evidence for variable-order fractional diffusion in lysosomes is presented. The final chapter presents a neural network, trained on fractional Brownian motion, which estimates the Hurst exponent of a trajectory using as few as seven data points. This neural network is then applied using a windowed approach to trajectories of GFP-Rab5 tagged endosomes, GFP-SNX1 tagged endosomes and lysosomes. The Hurst exponents estimated locally in time for single experimental trajectories form distinct probability density functions for each different intracellular vesicle and various statistics on the vesicle dynamics are extracted. In doing so, the potential of this new technique is demonstrated.