A Hamilton cycle is one of the most basic global structures one might wish to find in a graph.  It is a cycle that visits every vertex of the graph exactly once, making it a structure of interest in problems such as the famous Traveling Salesman Problem.  Determining whether a given graph has a Hamilton cycle is, in general, an NP-complete problem.  Consequently, a significant body of research has focused on identifying conditions that guaranteethe existence of Hamilton cycles.  In this talk, I will discuss several key results connecting Hamiltonicity to degree conditions, connectivity, and spectral properties of a graph. I will conclude by presenting our recent resolution of Jackson’s conjecture from 1981, which states that every d-regular oriented graph on n vertices has a Hamilton cycle, provided that d is at least n/4.