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Archaeological data provide a potential to investigate the diffusion of technological and cultural traits. However, much of this research agenda currently needs more formal quantitative methods to address small sample sizes and chronological uncertainty. This paper introduces a novel Bayesian framework for inferring the shape of diffusion curves using radiocarbon data associated with the presence/absence of a particular innovation. We developed two distinct approaches: 1) a hierarchical model that enables the fitting of an s-shaped diffusion curve whilst accounting for inter-site variations in the probability of sampling the innovation itself, and 2) a non-parametric model that can estimate the changing proportion of the innovation across user-defined time-blocks. The robustness of the two approaches was first tested against simulated datasets and then applied to investigate three case studies, the first pair on the diffusion of farming in prehistoric Japan and Britain and the third on cycles of changes in the burial practices of later prehistoric Britain. • Bayesian methods for modelling diffusion curves with radiocarbon dates are introduced. • Sample interdependence and non-sigmoid diffusion curves can be handled. • Robustness is tested on simulated datasets and archaeological case studies. • Code and data are available on a GitHub repository and archived on Zenodo. [ABSTRACT FROM AUTHOR]