This paper deals with studying the current research addressing the issues of assessment in translation didactics. The critical analysis of relevant scientific resources is aimed at finding out important theoretical achievements, empirical contributions and best practices in the sphere of organization and realization of the assessment aspects in translation teaching for grounding theorizing of the assessment system in teaching translation in the realm of education. In particular, the paper focuses on approaches, aims, functions, objects, kinds, methods and instruments, their peculiarities and requirements to them in terms of kind of translation, language pair, subject domain, higher education grade level. The outlined issues include incorporating industry assessment practices into education and academic training; forms and peculiarities of feedback and evaluation functions; methods (holistic, analytical, combined) of measuring and assessing translation quality, level of competence in translation and its components, learning objectives, efficiency of the translation process; methods of translation process monitoring; assessment criteria; finding, analysis and classification of translator's errors and mistakes; classification of kinds of assessment and ways of their realization; peculiarities of formative and summative assessment in translation teaching; involving students in assessment procedure; self-assessment and peer assessment and ways of their efficiency providing; testing in translation teaching.