Yes, running shoes should have some degree of flexibility. This allows for a more natural and efficient foot movement during the running stride, promoting a smoother transition from heel to toe. However, the level of flexibility can vary based on the type of running shoe and individual preferences. Some runners prefer a more flexible shoe for a minimalist feel, while others require more stability and support. It's essential to find a balance of flexibility that suits your running style, providing the necessary comfort and performance without compromising on support and protection.