The subject of my dissertation is the structure of continua and, in particular, of physical space and time. Consider the region of space you occupy: is it composed of indivisible parts? Are the indivisible parts, if any, extended? Are there infinitesimal parts? The standard view that space is composed of unextended points faces both a priori and empirical difficulties. In my dissertation, I develop and evaluate several novel approaches to these questions based on metaphysical, mathematical and physical considerations. In particular, I develop and evaluate two infinitesimal theories of space based on Robinson's nonstandard analysis. I argue that Infinitesimal Gunk, according to which every region is further divisible and some regions have infinitesimal sizes, has distinct advantages over alternative gunky views. I also advance a new account of distance for atomistic space, the mixed account, in response to Weyl's tile argument, which is an influential argument against the view that space is composed of indivisible regions. Having these theories in stock, we make progress in discovering the best theory of continua.