Consider the representations of a Lusztig quantum group at a root of unity. The goal of this talk is to compute the dimensions of all higher Ext spaces between two irreducibles. If the irreducibles belong to a regular block (which will be defined in the talk), it is known for a long time that the Kazhdan-Lusztig polynomials provide the answer. For singular blocks, translation functors and a positive grading (Koszulity) enable us to use the formulas in a regular block to obtain the Ext formulas we want. An important observation here is that the Ext^n space vanishes in every other degree. Much of the talk will consist of basic theories for the representations of root of unity quantum groups.