The Graceful Tree Conjecture was published almost 40 years ago and nowadays it is still an open problem. It seems as difficult to solve as it is to enunciate: every tree admits a vertex labeling on 0, 1, ..., n-1 such that the set of values of the differences of the numbers assigned to the ends of each edge is the set {1, 2, ..., n-1}. This Thesis has two major parts. The first one is a review of the most remarkable advances towards the solution of the problem. The second one is a self-developed method to gracefully label trees using adjacency matrices.