This paper is aimed at providing a comparative overview and critical analysis of the most important approaches to the assessment of student academic achievements, first and foremost, the classic psychometric and authentic assessment. The following are discussed: philosophical/epistemological viewpoint, nature of knowledge, educational discourse, assessment procedures, as well as potential possibilities and limitations of the abovementioned approaches. We provide a concise overview of the most important conceptual and methodological changes in the field of assessment of student academic achievements during the last decades of the 20th century. The contemporary view of assessment implies inevitable changes in the assessment approaches and practices hitherto, shifting the focus to the very process and context of assessment, as well as to self-assessment as a part of learning process. Instead of the usual juxtaposition of the most important approaches to assessment of student achievements, we argue that each of them, under certain circumstances, can contribute to the formation of the image about the student and his/her potentials which would be as objective and comprehensive as possible.