In this paper we review the topic of attosecond electron dynamics in molecular systems. We present a digest of recent research on this topic conducted by ourselves and other researchers with the intention of providing an accessible, but rigorous, account of the current state of this intriguing field of research. A short account of the background quantum theory is given before discussing recent theoretical advances on understanding correlation driven electron dynamics and electron nuclear coupling in molecules undergoing fast photoionization. We then review experimental advances, using both high harmonic generation and XFEL based ultrafast x-ray pulses, and provide three recent case studies from our own work to illustrate this. The final sections look forward to the next steps in this field: we discuss the prospect for controlling attochemistry as well as extending attosecond measurement methods to electron dynamics in larger molecules and condensed phase systems.