Smectite clay minerals and their suspensions have long been of both great scientific and applications interest and continue to display a remarkable range of new and interesting behaviour. Recently there has been an increasing interest in the properties of mixed suspensions of such clays with nanoparticles of different size, shape and charge. This review aims to summarize the current status of research in this area focusing on phase behaviour and rheological properties. We will emphasize the rich range of data that has emerged for these systems and the challenges they present for future investigations. The review starts with a brief overview of the behaviour and current understanding of pure smectite clays and their suspensions. We then cover the work on smectite clay-inorganic nanoparticle mixed suspensions according to the shape and charge of the nanoparticles - spheres, rods and plates either positively or negatively charged. We conclude with a summary of the overarching trends that emerge from these studies and indicate where gaps in our understanding need further research for better understanding the underlying chemistry and physics.