This paper synthesises the existing research on the dynamics of innovation diffusion, with a focus on Bass-type models and their extensions. The theoretical foundation of innovation diffusion proposed by Rogers (1962) and the seminal work of Bass (1969) serve as a starting point for the analysis. We identify and examine various generalizations and stochastic extensions of the Bass model, including counting processes, diffusion processes, and uncertain processes, as well as parameter estimation techniques, from classical statistical techniques to more advanced techniques such as Bayesian filtering and metaheuristic optimisation. We finally explore alternative models of innovation diffusion, with a particular focus on agent-based models. This overview of the evolution of Bass-type models illustrates the progress made in innovation diffusion research over the past decades.