For many classical algorithmic graph problems, faster algorithms are known for graphs that have additional structure. This short course will survey some important algorithmic techniques for graphs that can be drawn in the plane or other surfaces without crossing edges. The course will introduce several fundamental mathematical tools, including Euler's formula, rotation systems, duality, tree-cotree decompositions, the combinatorial Gauss-Bonnet theorem, homotopy, homology, covering spaces, balanced separators, and treewidth, as well as applications of these tools for computing minimum spanning trees, shortest paths, minimum cuts, and approximation solutions for several NP-hard prob