Bicycles are among the simplest and at the same time most fascinating vehicles. Despite their apparently basic design, their stability, dynamics, performance, and safety are still largely unknown. In a world where cities suffer from critical air pollution levels and are often congested by car traffic, bicycles can represent a cost-effective and ready solution to address part of the problem. However, the increasing use of bicycles enlarges the risk of bicycle-related accidents and injuries. Many of these accidents involve falls, highlighting the need of proper studies on bicycle-rider dynamics. While research has been conducted on the effect of bicycle parameters on the handling quality properties, we still do not know enough about the specific role of tyres on bicycle-rider dynamics. To better simulate what happens in the real world, existing bicycle mathematical models need to be integrated with robust tyre models that include side slip, as they can change both the lateral and longitudinal dynamics. Actually, the use of nonlinear tyre models allows taking into account the lateral force and self-aligning torque saturation. This is needed for instance to model the wobble as a non-divergent vibration modes, contrarily to what happens when using a simple linear tyre model. These features relate directly to predict critical situations for which we need a deep knowledge of bicycle tyres, thus proper test-rigs to measure their mechanical characteristics...