This paper presents the principles of operation of Resistive AC-Coupled Silicon Detectors (RSDs) and measurements of the temporal and spatial resolutions using a combined analysis of laser and beam test data. RSDs are a new type of n-in-p silicon sensor based on the Low-Gain Avalanche Diode (LGAD) technology, where the n+ implant has been designed to be resistive, and the read-out is obtained via AC-coupling. The truly innovative feature of RSD is that the signal generated by an impinging particle is shared isotropically among multiple read-out pads without the need for floating electrodes or an external magnetic field. Careful tuning of the coupling oxide thickness and the n+ doping profile is at the basis of the successful functioning of this device. Several RSD matrices with different pad width-pitch geometries have been extensively tested with a laser setup in the Laboratory for Innovative Silicon Sensors in Torino, while a smaller set of devices have been tested at the Fermilab Test Beam Facility with a 120 GeV/c proton beam. The measured spatial resolution ranges between 2.5μm for 70–100 pad-pitch geometry and 17μm with 200–500 matrices, a factor of 10 better than what is achievable in binary read-out (binsize∕12). Beam test data show a temporal resolution of ∼40ps for 200 μm pitch devices, in line with the best performances of LGAD sensors at the same gain.