This dissertation is an examination of the foundations of what I call a "fourth" tradition of analytical-scientific philosophy, the tradition of "structuralism". It is a disavowal of a metaphysics of substances and/or entities in the pursuit of scientific theory and truth. We look, in particular, at the current manifestation of this tradition, which advances the thesis of "structural realism"; we ask how tenable this thesis is, and whether we can weaken it. I argue that we should focus on methodology--a program for the formulation of scientific hypotheses about the sorts of things there are--rather than on metaphysics per se. We replace "substance" with "relation" as the basic ontic posit, and hold that substances or entities are metaphysically derivative from relational structure. Thus, the thesis is not that "there are no things" (or that "everything must go", as Ladyman et al. suppose); rather, the thesis is that the things (entities or substances) are relational structure, and there is no complete specification of an independent entity that is not itself more relational structure (so a metaphysics of substances is merely secondary to that of relational structure). I also suggest that there is no complete, unitary or monistic theory of what `structure' itself is. That is, I hold that there is no "total" structure of which everything that is relational structure is a "part", on the grounds that this would constitute an "illegitimate totality" in Russell's sense (the claim that "everything is structural" does not mean that there is a single structure which everything has--what a monistic theory of structure seems to demand). We then turn to the question of scientific explanation in light of structural realism: can there be explanation without a metaphysics of substances? I answer affirmatively. I then turn to two cases where, I argue, structuralism (and the specific thesis of structural realism) is in play regarding scientific explanation: quantum information theory, and the recent attempt to render quantum mechanics local by re-interpreting physical law time symmetrically. I conclude with a consideration of some objections to structuralism, and an articulation of the general view of metaphysics that structuralism seems to presuppose.