In this talk, I will discuss two different ‘ranking’ models: Mallows’ ranking model and rank-dependent diffusions. In the first part, I will discuss the rank-dependent diffusions. I will focus on two models: Up the River model, and N-player games with fuel constraints. These problems require treating carefully the corresponding PDEs. The former is joint with Li-Cheng Tsai, and the latter joint with Xin Guo and Renyuan Xu.  In the second part, I will focus on the Mallows’ permutation, and various Generalizations. In particular, I will talk about a general model, called regenerative permutations. I will also discuss the statistical properties and algorithms of these Mallows’ type Ranking models. This is partly joint with Jim Pitman.  If time permits, I will discuss recent progress on the random walk derived from random permutations, which is motivated by applications in systems biology.