Our research breaks down into three parts. First, we are exploring the mathematical foundations of program specifications and applying the understanding that we gain to the subject of specification language design. The success of a specification and testing system hinges on the quality of the underlying specification language. We are particularly concerned with the special problems posed by floating point arithmetic and concurrency. Second, we are developing more sophisticated methods - employing a judicious combination of verification and testing - for certifying that programs implement their formal specifications. Finally, to test the effectiveness of ideas, we are building an experimental program specification and testing system as an extension of the R^n Programming Environment already under development at Rice.