Healthcare cognitive assistants (HCAs) are intelligent systems or agents that interact with users in a context-aware and adaptive manner to improve their health outcomes by augmenting their cognitive abilities or complementing a cognitive impairment. They assist a wide variety of users ranging from patients to their healthcare providers (e.g., general practitioner, specialist, surgeon) in several situations (e.g., remote patient monitoring, emergency response, robotic surgery). While HCAs are critical to ensure personalized, scalable, and efficient healthcare, there exists a knowledge gap in finding the emerging trends, key challenges, design guidelines, and state-of-the-art technologies suitable for developing HCAs. This survey aims to bridge this gap for researchers from multiple domains, including but not limited to cyber-physical systems, artificial intelligence, human-computer interaction, robotics, and smart health. It provides a comprehensive definition of HCAs and outlines a novel, practical categorization of existing HCAs according to their target user role and the underlying application goals. This survey summarizes and assorts existing HCAs based on their characteristic features (i.e., interactive, context-aware, and adaptive) and enabling technological aspects (i.e., sensing, actuation, control, and computation). Finally, it identifies critical research questions and design recommendations to accelerate the development of the next generation of cognitive assistants for healthcare.