The unexpected spread of COVID-19 (SARS-CoV-2) has had a devastating impact worldwide. The measures taken to tackle it have affected economic, political, and social decisions. Likewise, the effects on public organizations have required organizational changes and solutions to respond to the demands and fulfillment of objectives during this critical period. A determining factor in times of organizational change and uncertainty is the leadership role that supports the achievement of organizational objectives in an effective and efficient manner in the face of any unexpected event. Therefore, it is necessary to focus on identifying the leader competencies required for public management in times of crisis. This paper aims to perform a systematic literature review based on the PRISMA approach (n = 25) to map current studies related to leadership in the public sector and to build a preliminary theoretical framework on the competencies observed in public leaders to overcome the current crisis and that must persist in the face of the new normal and future challenges that will test the adaptive capacity of the public sector.