The classical Borsuk-Ulam theorem has numerous consequences across mathematics. It states that any continuous map from an n-sphere to R^n must identify antipodal points. I will present some new applications of this result, such as:
1. Codes in projective spaces give bounds for the Gromov--Hausdorff distance between spheres of different dimensions.
2. For which pairs (d,n) does a continuous fibration of a region in R^n by unit d-spheres exist?
3. Generalizations of Lovász' lower bounds for chromatic numbers of graphs to obstructions for chromatic mixing.