Planar antennas have become an integral component in modern biomedical instruments owing to their compact structure, cost effectiveness, and light weight. These antennas are crucial in realizing medical systems such as body area networks, remote health monitoring, and microwave imaging systems. Antennas intended for the above applications should be conformal and fabricated using lightweight materials that are suitable for wear on the human body. Wearable antennas are intended to be placed on the human body to examine its health conditions. Hence, the performance of the antenna, such as its radiation characteristics across the operating frequency bands, should not be affected by human body proximity. This is achieved by selecting appropriate conformal materials whose characteristics remain stable under all environmental conditions. This paper aims to highlight the effects of human body proximity on wearable antenna performance. Additionally, this paper reviews the various types of flexible antennas proposed for biomedical applications. It describes the challenges in designing wearable antennas, the selection of a flexible material that is suitable for fabricating wearable antennas, and the relevant methods of fabrication. This paper also highlights the future directions in this rapidly growing field. Flexible antennas are the keystone for implementing next-generation wireless communication devices for health monitoring and health safety applications.