This chapter introduces a more abstract definition of smooth manifolds using charts and maximal atlases. We define the tangent space of an abstract smooth manifold at a point via equivalence classes of curves, establish its vector space structure, and construct the derivative of a smooth map. We show that our earlier, more concrete definitions align naturally with this abstract framework. Several examples are explored, including the circle, torus, Klein bottle, real projective spaces, and Grassmannians. We introduce the canonical bundle on Grassmannians and demonstrate that the canonical line bundle on real projective space is non-trivial. Exercises include the study of Hopf manifolds and Milnor hypersurfaces.