We pr esent a mechanics-informed machine learning framework for the data-driven constitutive modeling of nonlinearly elastic and viscoelastic materials. By design, it forces the architecture of a neural network to satisfy a list of hard constraints, including: dynamic stability, material stability, and internal variable stability; objectivity; consistency; fading memory; recovery of elasticity; and the 2nd law of thermodynamics. We show that e mbedding these notions in a learning approach reduces sensitivity to noise and promotes robustness to inputs outside the training domain.