There are merely no books on a detailed description of singularity formation for nonlinear evolution partial differential equations (PDE), especially concerning the construction and stability of blowup solutions, despite the fascinating aspect of this topic. This course is based on the instructor's research theme and aims at explaining briefly some key concepts, with a level of difficulty accessible to graduate students. The objective is to show how, despite the diversity of the PDE world, there are universal features in singularity formation. In particular, we focus on the self-similarity structures and their stability for the two classical parabolic problems: the semilinear heat equation and the Keller-Segel system modeling chemotaxis in biology. The course will also introduce symbolic Math Toolbox for solving PDEs.