The purpose of my talk will be to present some of the results that have been established for Littlewood’s conjecture. In the first part of the discussion I will give an expository overview of what is currently known about the conjecture and what are some of the questions that naturally arise from it. In the second part I will present in more detail the seminal work of Andrew Pollington and Sanju Velani from their 2001 paper: “On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture.”