The classical theory-driven approach to describe the deformation of a material body is based on the formulation of constitutive equations relating strains and stresses. A drawback of this approach is the effort typically required to develop appropriate functional relations and to identify material parameters. These efforts are not required in data-driven approaches to constitutive modeling. To combine the advantages and overcome the disadvantages of both theory-driven and data-driven constitutive modeling, we have developed the novel concept of Constitutive Artificial Neural Networks (CANNs). This machine learning approach to data-driven constitutive modeling does not require any major a priori assumptions about the constitutive law, but still incorporates substantial theoretical knowledge of continuum mechanics and constitutive theory. In this way, CANNs are able to learn the constitutive law of a material from relatively small amounts of stress-strain data. Moreover, our results suggest that CANNs can robustly discover different flavors of material models from data and, by design, have a clear physical interpretation. These capabilities are illustrated using different data collected from arterial and brain tissue.