A differentiable manifold, defined with the help of collections of charts, comes with basic notions of calculus prior to the introduction of the metric. The most important is how one may define vectors as directional derivatives, 1-forms via their dual pair, and build up general tensors from these two. Partial derivatives for functions extend to the Lie derivative, while a special subclass of tensors known as differential forms admits another type of derivative known as the exterior derivative. We develop calculus based on these most basic differential structures, ending with the Stokes theorem. Once a differentiable manifold is given, one can turn to a further structure of a metric and its accompanying Levi-Civita connection, leading to the Riemann curvature tensor. Numerous additional structures made possible by the covariantly constant metric are introduced, such as raising and lowering of indices, Killing vector fields, the volume form, the Hodge star map, geodesics, and geodesic normal coordinates.