Random vector functional-link (RVFL) networks are randomized multilayer perceptrons with a single hidden layer and a linear output layer, which can be trained by solving a linear modeling problem. In particular, they are generally trained using a closed-form solution of the (regularized) least-squares approach. This paper introduces several alternative strategies for performing full Bayesian inference (BI) of RVFL networks. Distinct from standard or classical approaches, our proposed Bayesian training algorithms allow to derive an entire probability distribution over the optimal output weights of the network, instead of a single pointwise estimate according to some given criterion (e.g., least-squares). This provides several known advantages, including the possibility of introducing additional prior knowledge in the training process, the availability of an uncertainty measure during the test phase, and the capability of automatically inferring hyper-parameters from given data. In this paper, two BI algorithms for regression are first proposed that, under some practical assumptions, can be implemented by a simple iterative process with closed-form computations. Simulation results show that one of the proposed algorithms, Bayesian RVFL, is able to outperform standard training algorithms for RVFL networks with a proper regularization factor selected carefully via a line search procedure. A general strategy based on variational inference is also presented, with an application to data modeling problems with noisy outputs or outliers. As we discuss in this paper, using recent advances in automatic differentiation this strategy can be applied to a wide range of additional situations in an immediate fashion.