Cutting-edge technologies have been widely employed in healthcare delivery, resulting in transformative advances and promising enhanced patient care, operational efficiency, and resource usage. However, the proliferation of networked devices and data-driven systems has created new cybersecurity threats that jeopardize the integrity, confidentiality, and availability of critical healthcare data. This review paper offers a comprehensive evaluation of the current state of cybersecurity in the context of smart healthcare, presenting a structured taxonomy of its existing cyber threats, mechanisms and essential roles. This study explored cybersecurity and smart healthcare systems (SHSs). It identified and discussed the most pressing cyber threats and attacks that SHSs face, including fake base stations, medjacking, and Sybil attacks. This study examined the security measures deployed to combat cyber threats and attacks in SHSs. These measures include cryptographic-based techniques, digital watermarking, digital steganography, and many others. Patient data protection, the prevention of data breaches, and the maintenance of SHS integrity and availability are some of the roles of cybersecurity in ensuring sustainable smart healthcare. The long-term viability of smart healthcare depends on the constant assessment of cyber risks that harm healthcare providers, patients, and professionals. This review aims to inform policymakers, healthcare practitioners, and technology stakeholders about the critical imperatives and best practices for fostering a secure and resilient smart healthcare ecosystem by synthesizing insights from multidisciplinary perspectives, such as cybersecurity, healthcare management, and sustainability research. Understanding the most recent cybersecurity measures is critical for controlling escalating cyber threats and attacks on SHSs and networks and encouraging intelligent healthcare delivery.