The next frontier towards truly ubiquitous connectivity is the use of Low Earth Orbit (LEO) small-satellite constellations to support 5G and Beyond-5G (B5G) networks. Besides enhanced mobile broadband (eMBB) and massive machine-type communications (mMTC), LEO constellations can support ultra-reliable communications (URC) with relaxed latency requirements of a few tens of milliseconds. Small-satellite impairments and the use of low orbits pose major challenges to the design and performance of these networks, but also open new innovation opportunities. This paper provides a comprehensive overview of the physical and logical links, along with the essential architectural and technological components that enable the full integration of LEO constellations into 5G and B5G systems. Furthermore, we characterize and compare each physical link category and explore novel techniques to maximize the achievable data rates.