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Software architectures are multi-dimensional entities that can be fully understood only when viewed and analyzed at four different levels of abstraction: (1) internal functionality of a component, (2) the interface(s) exported by the component to the rest of the system, (3) interconnection of architectural elements in an architecture, and (4) rules of the architectural style. This paper presents the characteristics of each of the four levels of architectural abstraction, outlines the kinds of analyses that need to be performed at each level, and discusses the kinds of formal notations that are suitable at each level. We use the pipe-and-filter and Chiron-2 (C2) architectural styles as illustrations. In particular, we present formal models of C2 at the last three levels of abstraction as a first step in enabling a C2 design environment to perform the necessary analyses of architectures. We discuss the benefits of the formal definitions and our experience to date.