Intracellular transport is driven primarily by molecular motor proteins, such as kinesin and dynein, which convert chemical energy (stored in ATP) to directed motion along microtubules. Mathematical models of this process typically involve stochastic elements to describe the progress of binding and other chemical steps in the molecular motor stepping cycle, as well as for the interaction of the spatial dynamics with thermal fluctuations in the environment. Experiments and models have been pursued for the last 20 years to explore the functioning and characteristics of single molecular motors attached to a cargo load. Attention has turned more recently to understand how intracellular transport operates in the presence of multiple motors of various types, which is evidently relevant in vivo. Of particular interest is understanding mechanistically at a systems level how the various molecular motors, microtubule architecture, and regulatory factors interact to achieve the functional goals of intracellular transport. Achieving this goal appears to require the integration of experimental data and theoretical models over at least three scales of description, and the development of quantitative relationships between in vitro and in vivo experimental results. I will sketch some of the emerging conceptual challenges and opportunities in the science of intracellular transport, and several ways in which stochastic models and methods have been employed by collaborators and other groups to contribute to our understanding of this complex system. My recent and ongoing research in this area is pursued jointly with John Fricks, Scott McKinley, Will Hancock, and Avanti Athreya.