  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 $2.5\; \mu m$ for 70-100 pad-pitch geometry
and $17\; \mu m$ with 200-500 matrices, a factor of 10 better than what is
achievable in binary read-out ($bin\; size/ \sqrt{12}$). Beam test data show a
temporal resolution of $\sim 40\; ps$ for 200-$\mu m$ pitch devices, in line
with the best performances of LGAD sensors at the same gain.
