Mathematicians have studied the interpolation problem for a long time. A well-known version, going back to Euclid himself, asks the following: given a finite set of points in the euclidean plane, is there a polynomial of prescribed degree in two variables having them as roots? A variant which is of interest to us requires the polynomial to vanish at the given points up to a certain “multiplicity”. We will first introduce this problem by looking at some examples and then discuss some techniques to study the problem. Later we will talk about connections with interesting recent work in commutative algebra and algebraic geometry, beginning with the famous “Nagata Conjecture”.