Today, 5G networks are being worldwide rolled out, with significant benefits in our economy and society. However, 5G systems alone are not expected to be sufficient for the challenges that 2030 networks will experience, including, e.g., always-on networks, 1 Tbps peak data rate, <10 cm positioning, etc. Thus, the definition of evolutions of the 5G systems and their (r)evolutions are already being addressed by the scientific and industrial communities, targeting 5G-Advanced (5G-A) and 6G. In this framework, Non-Terrestrial Networks (NTN) have successfully been integrated in 3GPP Rel. 17 and it is expected that they will play an even more pivotal role for 5G-A (up to Rel. 20) and 6G systems (beyond Rel. 20). In this paper, we explore the path that will lead to 5G-A and 6G NTN communications, providing a clear perspective in terms of system architecture, services, technologies, and standardisation roadmap.