This is a course for undergraduate and graduate mathematics and physics students. Its purpose is to give an introduction to the theory of manifolds -- topological spaces which locally look as the usual Euclidean space but may have a complicated global shape. Manifolds play a basic role in modern mathematics and mathematical physics.

We will cover the following topics:

1) Manifolds (definition, examples, constructions).
2) Topological properties of manifolds and partition of unity.
3) Tangent space and the differential.
4) Immersions, submersions, and submanifolds.
5) Embedding and Whitney's theorem.
6) Vector fields and flows (Lie bracket & Lie derivative).
7) Orientation. 
8) Degree of a map (and applications: Brouwer fixed point, Borsuk-Ulam, the "hairy ball" theorem, etc.).
9) Differential forms.
10) Integration on manifolds (Stokes theorem).
11) de Rham cohomology.
If time permits:
12) Vector bundles, parallel transport, connection and covariant differentiation, curvature.