From remote sensing of the environment, to brain scans in medicine, the growth in the use of image data has motivated a parallel increase in statistical techniques for analysing these images. A particular area of growth has been in Bayesian models and corresponding computational methods. Bayesian approaches have been proposed to address the gamut of supervised and unsupervised inferential aims in image analysis. In this article we provide a general review of these approaches, with a focus on unsupervised analysis of 2-D images. Four exemplar methods that canvas the broad aims of image modelling and analysis are described. An exposition of these approaches is provided by applying them to an environmental case study involving the use of satellite data to assess water quality in the Great Barrier Reef, Australia. The techniques considered in detail are hidden Markov random fields (MRF), Gaussian MRF, Poisson/gamma random fields, and Voronoi tessellations. We also consider a variety of enabling computational algorithms, including MCMC, variational Bayes and integrated nested Laplace approximations. We compare the different aims and inferential capabilities of the models and discuss the advantages and drawbacks of the corresponding computational algorithms.