A scalable programming language is one in which the same concepts can describe small as well as large parts. Towards this goal, Scala unifies concepts from object and module systems. In particular, objects can contain type members, which can be selected as types, called path-dependent types. Focusing on path-dependent types, we develop a type-theoretic foundation for Scala: the calculus of Dependent Object Types (DOT). We derive DOT from System F<:, in small steps: (1) in Translucent F<:>, we add a lower bound to each type variable, in addition to its usual upper bound, (2) in System D, we turn each type variable into a regular term variable containing a type, (3) for a full subtyping lattice, we add intersection and union types, (4) for objects, we consolidate all values into records, (5) for objects that close over a self, we introduce a recursive type, binding a self term variable, (6) for recursive types, we first extend the theory in typing and then also in subtyping. Through this bottom-up exploration, we discover a sound, uniform yet powerful design for DOT. We devise strategies and techniques for proving soundness that scale through this iterative step-by-step process: (1) "pushback" of subtyping transitivity or subsumption, to concisely capture inversion of subtyping or typing, (2) distinction between concrete vs. abstract context variables, to resolve tension between preservation of types vs. preservation of type abstractions, (3) and, specifically for big-step semantics, a type that closes over an environment, to relate context-dependent types across closures. While ultimately, we have developed sound models of DOT in both big-step and small-step operational semantics, historically, the shift to big-step semantics has been helpful in focusing the requirements. In particular, by developing a novel big-step soundness proof for System F<:, calculi like System D<: emerge as straightforward generalizations, almost like removing artificial restrictions. Interesting in their own right, our type soundness techniques for definitional interpreters extend to mutable references without use of co-induction. The DOT calculus finally grounds languages like Scala in firm theory. The DOT calculus helps in finding bugs in Scala, and in understanding feature interaction better as well as requirements. The DOT calculus serves as a good basis for future work which studies extensions or encodings on top of the core, bridging the gap from DOT to Dotty / Scala.