Affine and convex spaces: blending the analytic and geometric viewpoints

This is a short introduction to affine and convex spaces, written especially for physics students. It summarizes different elementary presentations available in the mathematical literature, and blends analytic- and geometric-flavoured presentations. References are also provided, as well as a brief discussion of Grassmann spaces and an example showing the relevance and usefulness of affine spaces in Newtonian physics.