Polymers are one of the most popular materials for next-generation flexible sensing device fabrication due to their tunable mechanical and electrical properties. A series of prior research studies in the field of smart flexible and wearable sensing illustrates the potential of various polymer and composite materials to be applied in sensor development. In this direction, mechanical compliance plays a vital role as it ensures the stability and reliability of the fabricated sensor. Therefore, engineering mechanical compliance for the development of smart flexible solutions has emerged as a significant area of research. Furthermore, the usage of flexible sensing devices is rapidly increasing in the field of healthcare devices and robotic automation. This feature article summarizes the relevant contributions of the authors in the field of engineered polymers and composites for flexible sensor development with a focus on healthcare and physical sensing applications. We discuss the polymer and composite materials, their characteristics, fabrication technologies, finite element method analysis, and examples of flexible physical sensors, i.e. pressure, strain, and temperature sensors, for various wearable healthcare applications and robotic automation. Finally, we discuss examples of multi-sensory systems having flexible sensors.