Experiments where the response of interest is a curve or 'profile' arise in a variety of applications in engineering practice. In a recent paper (Journal of Quality Technology, 44, 2, pp. 117–135, 2012), a mixed-effects Bayesian approach was proposed for the Bayesian optimization of profile response systems, where a particular shape of the profile response defines desired properties of the product or process. This paper proposes an alternative spatio-temporal Gaussian random function process model for such profile response systems, which is more flexible with respect to the types of desired profile shapes that can be modeled and allows us to model profile-to-profile correlation, if this exists. The method is illustrated with real examples taken from the literature, and practical aspects related to model building and diagnostics are discussed. Copyright © 2013 John Wiley & Sons, Ltd.