6G networks can support global, ubiquitous and seamless connectivity through the convergence of terrestrial and non-terrestrial networks (NTNs). Unlike terrestrial scenarios, NTNs pose unique challenges including propagation characteristics, latency and mobility, owing to the operations in spaceborne and airborne platforms. To overcome all these technical hurdles, this survey paper presents the use of artificial intelligence (AI) techniques in learning and adapting to the complex NTN environments. We begin by providing an overview of NTNs in the context of 6G, highlighting the potential security and privacy issues. Next, we review the existing AI methods adopted for 6G NTN optimization, starting from machine learning (ML), through deep learning (DL) to deep reinforcement learning (DRL). All these AI techniques have paved the way towards more intelligent network planning, resource allocation (RA), and interference management. Furthermore, we discuss the challenges and opportunities in AI-powered NTN for 6G networks. Finally, we conclude by providing insights and recommendations on the key enabling technologies for future AI-powered 6G NTNs.