At the International Congress of Mathematicians in 1900 in Paris, Hilbert presented his now famous list of 23 problems. Many of these were rather influential for 20th century mathematics. One of these is Hilbert's 17th problem: Given a real polynomial in several variables that is non-negative, can it be represented as a sum of squares of rational functions? Artin answered this questions in 1927 in the affirmative. In this talk we will take a closer look at both Hilbert's problem and Artin's solution, and try to understand how all this relates to the upcoming lecture of M. F. Roy.