In 2007 we celebrate the 300th anniversary of Euler's birth. Many aspects of Euler's vast output will be examined during this year. In this talk we focus on Euler's research in Diophantine problems, which were a long-term interest of Euler and are still of interest today. We consider particular highlights from Euler's work on Diophantine equations, such as his landmark text Algebra, his work on Fermat's Last Theorem, and the Euler conjecture. Many of the problems considered by Euler fall under the heading of what are now called Euler sums, which are Diophantine equations equating sums of like powers. For example, in a paper from 1754 we see Euler discussing the problem of when the sum of three cubics will equal a cubic. We examine the subsequent history of research on Euler sums.