Large Language Models (LLMs) have accelerated the emergence of agentic Artificial Intelligence (AI) systems in healthcare, representing a paradigm shift from static models to autonomous entities capable of complex medical reasoning. This survey examines AI agents across six domains: clinical decision support, healthcare systems management, clinical consultation, medical education, medical imaging, and evaluation frameworks. We analyze over 30 systems including MDAgents, Agent Hospital, ColaCare, and AgentClinic, identifying key architectural patterns and performance characteristics. These systems demonstrate promising results across a range of clinical tasks, leveraging both multi-agent collaboration and specialized medical reasoning. Critical deployment barriers include data privacy, algorithmic bias, regulatory gaps, safety concerns in multi-agent architectures, explainability requirements, and integration challenges. We identify open research problems and highlight key challenges to advancing safe, reliable, and impartial healthcare agentic AI.