This is a review paper in which different definitions of achievement are analyzed and different possibilities for test construction are explored. A first characterization of achievement is accomplished through the analysis of construct representation. From this perspective, the behavioral approach focuses more on the end result, whereas the cognitive approach is more process centered. In a second stage, this review analyzes the data about nomothetic amplitude: the relationships between achievement and aptitudes, socioeconomic status, and changes over time. The final section offers a view of the possibilities and difficulties involved in the attempt to substitute traditional methods for performance assessment methods. Given the difficulties and cost in development time, scoring and other variables, the review concludes by assigning a major role to computer technology in assessment, if performance assessment is going to have a chance to achieve widespread use. Keywords: achievement test, item response theory, automatic item generation, automatic scoring, performance assessment and test construction.