Let us consider the deconvolution problem, that is, to recover a latent source $x(\cdot)$ from the observations $\mathbf{y} = [y_1,\ldots,y_N]$ of a convolution process $y = x\star h + η$, where $η$ is an additive noise, the observations in $\mathbf{y}$ might have missing parts with respect to $y$, and the filter $h$ could be unknown. We propose a novel strategy to address this task when $x$ is a continuous-time signal: we adopt a Gaussian process (GP) prior on the source $x$, which allows for closed-form Bayesian nonparametric deconvolution. We first analyse the direct model to establish the conditions under which the model is well defined. Then, we turn to the inverse problem, where we study i) some necessary conditions under which Bayesian deconvolution is feasible, and ii) to which extent the filter $h$ can be learnt from data or approximated for the blind deconvolution case. The proposed approach, termed Gaussian process deconvolution (GPDC) is compared to other deconvolution methods conceptually, via illustrative examples, and using real-world datasets.