Abstract We propose a notion of orbit for bilinear mappings on Banach spaces. This then leads us to the concept of bihypercyclic bilinear mappings, that is, of bilinear mappings with a dense orbit. We provide some examples of such mappings, and we obtain some basic properties of bihypercyclicity. In our main result we show that every separable (real or complex) Banach space supports a bihypercyclic bilinear mapping. We also formulate some open problems.