Double neutron star (DNS) systems represent extreme physical objects and the endpoint of an exotic journey of stellar evolution and binary interactions. Large numbers of DNS systems and their mergers are anticipated to be discovered using the Square-Kilometre-Array searching for radio pulsars and high-frequency gravitational wave detectors (LIGO/VIRGO), respectively. Here we discuss all key properties of DNS systems, as well as selection effects, and combine the latest observational data with new theoretical progress on various physical processes with the aim of advancing our knowledge on their formation. We examine key interactions of their progenitor systems and evaluate their accretion history during the high-mass X-ray binary stage, the common envelope phase and the subsequent Case BB mass transfer, and argue that the first-formed NSs have accreted at most $sim 0.02;M_{odot}$. We investigate DNS masses, spins and velocities, and in particular correlations between spin period, orbital period and eccentricity. Numerous Monte Carlo simulations of the second supernova (SN) events are performed to extrapolate pre-SN stellar properties and probe the explosions. All known close-orbit DNS systems are consistent with ultra-stripped exploding stars. Although their resulting NS kicks are often small, we demonstrate a large spread in kick magnitudes which may, in general, depend on the past interaction history of the exploding star and thus correlate with the NS mass. We analyze and discuss NS kick directions based on our SN simulations. Finally, we discuss the terminal evolution of close-orbit DNS systems until they merge and possibly produce a short $gamma$-ray burst.
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