Cantor sets and their fractional dimensions are related to many unsolved conjectures. One of these conjectures, known as Palis' Conjecture, involves the dimension of the Minkowski sums of Cantor sets, and in this talk we will describe the details of the Palis Conjecture. In addition, we will use some of the tools involving Cantor sets to describe of the some properties of the Lagrange and Markov spectra, in particular, related to existence of the Hall's ray. The talk will be mostly based on the work done by Moreira in recent years.