This paper presents a review of Tactile Sensing Systems, with an emphasis on their application to the non-invasive characterization of tumors. Emulating the perceptual mechanism of the human skin, the Tactile Sensing Systems characterize tumors using touch sensors by quantifying the mechanical properties such as size and elasticity. The authors survey several tactile transduction methods: capacitive, piezoresistive, piezoelectric, magnetic, and optical. The advantages and disadvantages of different tactile sensors are discussed. The complex human sense of touch is emulated using tactile sensor data, novel data processing algorithms, and near real-time interpretation in a human-readable format. Tactile Sensing Systems utilize tactile sensors and other subsystems to come up with accurate mechanical properties of the touched objects. We review Elasticity Determination Systems, which are a special case of Tactile Sensing Systems. These systems are based on capacitive sensors, piezoelectric sensors, elastography, and optical tactile sensors. Then the optical Tactile Sensing System is discussed in detail; architecture, sensing principle, and algorithms to compute a risk score. Moreover, a survey of multimodal Tactile Sensing Systems, which broaden the capabilities of existing tactile sensing systems, is presented. The paper concludes with discussions and future research directions.