Composition of software elements into assemblies (systems) is a fundamental aspect of software development. It is an important strength of formal mathematical specification that the descriptions of elements can be precisely composed into the descriptions of assemblies. Testing, on the other hand, is usually thought to be “non-compositional. ” Testing provides information about any executable software element, but testing descriptions have not been combined to describe assemblies of elements. The underlying reason for the compositional deficiency of testing is that tests are samples. When two elements are composed, the input samples (test points) for the first lead to an output sample, but it does not match the input test points of the second, following element. The current interest in software components and component-based software development (CBSD) provides an ideal context for investigating elements and assemblies. In CBSD, the elements (components) are analyzed without knowledge of the system(s) to be later assembled. A fundamental testing theory of component composition must use measured component properties (test results) to predict system properties. This paper proposes a testing-based theory of software component composition based on subdomains. It shows how to combine subdomain tests of components into testing predictions for arbitrarily complex assemblies formed by sequence, conditional, and iteration constructions. The basic construction of the theory applies to functional behavior, but the theory can also predict system non-functional properties from component subdomain tests. Compared to the alternative of actually building and testing a system, the theoretical predictions are computationally more efficient. The theory can also be described as an exercise in modeling. Components are replaced by abstractions derived from testing them, and these models are manipulated to model system behavior.