We introduce universal solution manifold network (USM-Net), a novel surrogate model, based on artificial neural networks (ANNs), which applies to differential problems whose solution depends on physical and geometrical parameters. We employ a mesh-less architecture, thus overcoming the limitations associated with image segmentation and mesh generation required by traditional discretization methods. Our method encodes geometrical variability through scalar landmarks, such as coordinates of points of interest. In biomedical applications, these landmarks can be inexpensively processed from clinical images. We present proof-of-concept results obtained with a data-driven loss function based on simulation data. Nonetheless, our framework is non-intrusive and modular, as we can modify the loss by considering additional constraints, thus leveraging available physical knowledge. Our approach also accommodates a universal coordinate system, which supports the USM-Net in learning the correspondence between points belonging to different geometries, boosting prediction accuracy on unobserved geometries. Finally, we present two numerical test cases in computational fluid dynamics involving variable Reynolds numbers as well as computational domains of variable shape. The results show that our method allows for inexpensive but accurate approximations of velocity and pressure, avoiding computationally expensive image segmentation, mesh generation, or re-training for every new instance of physical parameters and shape of the domain.