This course is the second part of a two semester introduction to topology. We will start with a few basic notions from category theory that will make some of the results easier to state and easier to understand. Next, we will learn a few advanced topics of general topology, including the Tietze extension theorem, the Tychonoff theorem, and Stone-Cech compactification. The majority of the course will cover basic topics and examples in algebraic topology. Algebraic topology translates difficult questions about spaces into algebraic questions that can be more readily answered. Topics will include the fundamental group and covering spaces, the Seifert-van Kampen theorem, and the classification of surfaces.