As we transition from the 5G epoch, a new horizon beckons with the advent of 6G, seeking a profound fusion with novel communication paradigms and emerging technological trends, bringing once-futuristic visions to life along with added technical intricacies. Although analytical models lay the foundations and offer systematic insights, we have recently witnessed a noticeable surge in research suggesting machine learning (ML) and artificial intelligence (AI) can efficiently deal with complex problems by complementing or replacing model-based approaches. The majority of data-driven wireless research leans heavily on discriminative AI (DAI) that requires vast real-world datasets. Unlike the DAI, Generative AI (GenAI) pertains to generative models (GMs) capable of discerning the underlying data distribution, patterns, and features of the input data. This makes GenAI a crucial asset in wireless domain wherein real-world data is often scarce, incomplete, costly to acquire, and hard to model or comprehend. With these appealing attributes, GenAI can replace or supplement DAI methods in various capacities. Accordingly, this combined tutorial-survey paper commences with preliminaries of 6G and wireless intelligence by outlining candidate 6G applications and services, presenting a taxonomy of state-of-the-art DAI models, exemplifying prominent DAI use cases, and elucidating the multifaceted ways through which GenAI enhances DAI. Subsequently, we present a tutorial on GMs by spotlighting seminal examples such as generative adversarial networks, variational autoencoders, flow-based GMs, diffusion-based GMs, generative transformers, large language models, autoregressive GMs, to name a few. Contrary to the prevailing belief that GenAI is a nascent trend, our exhaustive review of approximately 120 technical papers demonstrates the scope of research across core wireless research areas, including 1) physical layer design; 2) network optimization, organization, and management; 3) network traffic analytics; 4) cross-layer network security; and 5) localization & positioning. Furthermore, we outline the central role of GMs in pioneering areas of 6G network research, including semantic communications, integrated sensing and communications, THz communications, extremely large antenna arrays, near-field communications, digital twins, AI-generated content services, mobile edge computing and edge AI, adversarial ML, and trustworthy AI. Lastly, we shed light on the multifarious challenges ahead, suggesting potential strategies and promising remedies. Given its depth and breadth, we are confident that this tutorial-cum-survey will serve as a pivotal reference for researchers and professionals delving into this dynamic and promising domain.