We formulate f(Q, C) gravity and cosmology. Such a construction is based on the symmetric teleparallel geometry, but apart form the non-metricity scalar Q we incorporate in the Lagrangian the boundary term C of its difference from the standard Levi-Civita Ricci scalar R. We extract the general metric and affine connection field equations, we apply them at a cosmological framework, and adopting three different types of symmetric teleparallel affine connections we obtain the modified Friedmann equations. As we show, we acquire an effective dark -energy sector of geometrical origin, which can lead to interesting cosmological phenomenology. Additionally, we may obtain an effective interaction between matter and dark energy. Finally, examining a specific model, we show that we can obtain the usual thermal history of the universe, with the sequence of matter and dark -energy epochs, while the effective dark -energy equation -of -state parameter can be quintessence -like, phantom -like, or cross the phantom -divide during evolution.