The purpose of this study is to identify the problems of the quality assurance system of higher education in the Republic of Kazakhstan. To do this, the opinions of the stakeholders (students, teaching staff, employers, administrative, and managerial staff) in the process were screened. The data obtained were compared with the results of independent international ratings and other parameters. For screening, field studies were conducted using a questionnaire for students, teachers, non-academic employees of universities, and employers covering all regions of Kazakhstan in terms of quality assurance components. Based on the constructivist approach, the answers were interpreted to identify the problems of the quality assurance system in the understanding of the study participants. This study revealed that participants rate the quality of the content at a below-average level, the quality of the contingent at an average level, the quality of staff at a lower-than-average level, and the quality of infrastructure at a low level. Comparing the results of the survey and interviewing with the analysis of the results of entrance exams and ratings made it possible to identify the most pressing problems in ensuring the quality of educational programs, contingent, staff, and infrastructure of universities and draw conclusions about its improvement. The obtained results will help to customize the system of quality assurance of higher and postgraduate education, thus contributing to the training of highly competitive specialists for the country’s economy.