Although convolutional neural networks (CNNs) showed remarkable results in many vision tasks, they are still strained by simple yet challenging visual reasoning problems. Inspired by the recent success of the Transformer network in computer vision, in this paper, we introduce the Recurrent Vision Transformer (RViT) model. Thanks to the impact of recurrent connections and spatial attention in reasoning tasks, this network achieves competitive results on the same-different visual reasoning problems from the SVRT dataset. The weight-sharing both in spatial and depth dimensions regularizes the model, allowing it to learn using far fewer free parameters, using only 28k training samples. A comprehensive ablation study confirms the importance of a hybrid CNN + Transformer architecture and the role of the feedback connections, which iteratively refine the internal representation until a stable prediction is obtained. In the end, this study can lay the basis for a deeper understanding of the role of attention and recurrent connections for solving visual abstract reasoning tasks. The code for reproducing our results is publicly available here: https://tinyurl.com/recvit.