In the context of state administration institutions, including institutions of direct administration, there are increasing attempts to improve their quality and efficiency. Thus, it is important that public institutions carry out their processes with quality, despite any political changes in state administration. The issue also relates to how often public institutions fulfil the planned indicators and with what accuracy they are fulfilled. Therefore, proper planning and use of resources is at question. The aim of the paper is to determine the place of a process and of the process approach in a public institution and the process approach’s ability to improve process management in a public institution and the operation of a public institution’s quality system. The research includes the application of the scope review method, systemic qualitative literature review method, Agile approach and impact-effort matrix. The paper first addresses the methods used for a theoretical understanding of process management approaches, process approach and processes, then continues to their analysis. Afterwards, the paper assesses the possible solutions in a test institution and through a test process, including by providing universally applicable proposals for improving the quality and efficiency of process management and functioning of a quality system in a public institution.