Imagine Genghis Khan, Aretha Franklin, and the Cleveland Cavaliers performing an opera on Maui. This silly sentence makes a serious point: As humans, we can flexibly generate and comprehend an unbounded number of complex ideas. Little is known, however, about how our brains accomplish this. Here we assemble clues from disparate areas of cognitive neuroscience, integrating recent research on language, memory, episodic simulation, and computational models of high-level cognition. Our review is framed by Fodor's classic language of thought hypothesis, according to which our minds employ an amodal, language-like system for combining and recombining simple concepts to form more complex thoughts. Here, we highlight emerging work on combinatorial processes in the brain and consider this work's relation to the language of thought. We review evidence for distinct, but complementary, contributions of map-like representations in subregions of the default mode network and sentence-like representations of conceptual relations in regions of the temporal and prefrontal cortex.