This paper presents a new remote healthcare model, which, exploiting wireless biomedical sensors, an embedded local unit (gateway) for sensor data acquisition-processing-communication, and a remote e-Health service center, can be scaled in different telemedicine scenarios. The aim is avoiding hospitalization cost and long waiting lists for patients affected by chronic illness who need continuous and long-term monitoring of some vital parameters. In the "1:1" scenario, the patient has a set of biomedical sensors and a gateway to exchange data and healthcare protocols with the remote service center. In the "1:N" scenario the use of gateway and sensors is managed by a professional caregiver, e.g., assigned by the Public Health System to a number N of different patients. In the "point of care" scenario the patient, instead of being hospitalized, can take the needed measurements at a specific health corner, which is then connected to the remote e-Health center. A mix of commercially available sensors and new custom-designed ones is presented. The new custom-designed sensors range from a single-lead electrocardiograph for easy measurements taken by the patients at their home, to a multi-channel biomedical integrated circuit for acquisition of multi-channel bio signals, to a new motion sensor for patient posture estimation and fall detection. Experimental trials in real-world telemedicine applications assess the proposed system in terms of easy usability from patients, specialist and family doctors, and caregivers, in terms of scalability in different scenarios, and in terms of suitability for implementation of needed care plans.