Fano varieties are ‘atomic pieces’ of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and classification of Fano varieties, with an emphasis on three-dimensional Fano varieties with mild singularities called \(\mathbb {Q}\) -Fano threefolds. The classification of \(\mathbb {Q}\) -Fano threefolds has been open for several decades, but there has been significant recent progress. These advances combine computational algebraic geometry and large-scale data analysis with new ideas that originated in theoretical physics.