Non-terrestrial networks (NTNs) traditionally have certain limited applications. However, the recent technological advancements and manufacturing cost reduction opened up myriad applications of NTNs for 5G and beyond networks, especially when integrated into terrestrial networks (TNs). This article comprehensively surveys the evolution of NTNs highlighting their relevance to 5G networks and essentially, how it will play a pivotal role in the development of 6G ecosystem. We discuss important features of NTNs integration into TNs and the synergies by delving into the new range of services and use cases, various architectures, technological enablers, and higher layer aspects pertinent to NTNs integration. Moreover, we review the corresponding challenges arising from the technical peculiarities and the new approaches being adopted to develop efficient integrated ground-air-space (GAS) networks. Our survey further includes the major progress and outcomes from academic research as well as industrial efforts representing the main industrial trends, field trials, and prototyping towards the 6G networks.