Abstract Recent research in neuropsychology and cognitive neuroscience has shown that episodic memory and future thinking engage a common core brain network. Some have interpreted these results as showing that episodic memory and future thinking are two operations of a single cognitive system for mental time travel. However, some recent results cast doubt upon this tensed-content view, as they strongly suggest that a temporal content may not be essential for the engagement of this core brain network. This chapter reviews evidence both in favor of and against the tensed-content view, and suggests instead a different approach that can accommodate these discrepant results: the dynamic-structure view. According to this view, what explains the engagement of the core brain network is not that the mental representations are about time but rather that they unfold in time, i.e. that they have a dynamic structure.