Low Earth Orbit (LEO) Satellite Network (SatNet) with their mega-constellations are expected to play a key role in providing ubiquitous Internet and communications services in the future. LEO SatNets will provide wide-area coverage and support service availability, continuity, and scalability. To support the integration of SatNets and terrestrial Fifth Generation (5G) networks and beyond, the satellite communication industry has become increasingly involved with the 3rd Generation Partnership Project (3GPP) standardization activities for 5G. In this work, we review the 3GPP standardization activities for the integration of SatNets in 5G and beyond. The 3GPP use cases of SatNets are highlighted and potential requirements to realize them are summarized as well. The impacted areas of New Radio (NR) are discussed with some potential solutions. The foreseen requirements for the management and orchestration of SatNets within 5G are described. Future standardization directions are discussed to support the full integration of SatNets in Sixth Generation (6G) with the goal of ubiquitous global connectivity.