Several wearable devices for physiological and activity monitoring are found on the market, but most of them only allow spot measurements. However, the continuous detection of physiological parameters without any constriction in time or space would be useful in several fields such as healthcare, fitness, and work. This can be achieved with the application of textile technologies for sensorized garments, where the sensors are completely embedded in the fabric. The complete integration of sensors in the fabric leads to several manufacturing techniques that allow dealing with both the technological challenges entailed by the physiological parameters under investigation, and the basic requirements of a garment such as perspiration, washability, and comfort. This review is intended to provide a detailed description of the textile technologies in terms of materials and manufacturing processes employed in the production of sensorized fabrics. The focus is pointed at the technical challenges and the advanced solutions introduced with respect to conventional sensors for recording different physiological parameters, and some interesting textile implementations for the acquisition of biopotentials, respiratory parameters, temperature and sweat are proposed. In the last section, an overview of the main garments on the market is depicted, also exploring some relevant projects under development.