In this thesis we explore graph decompositions under different constraints. The title of the is due to the fact that most of these decompositions are neighbour-distinguishing. That is, we can extract from each such decomposition a proper vertex colouring. Moreover, most of the considered decompositions are edge partitions, and therefore can be seen as edge-colourings. The main question presented in this thesis is was introduced by Karoński, Łuczak and Thomason in [KLT04]: Can we weight the edges of a graph G, with weights 1, 2, and 3, such that any two of adjacent vertices of G are distinguished by the sum of their incident weights ? This question later becomes the famous 1-2-3 Conjecture. In this thesis we explore several variants of the 1-2-3 Conjecture, and their links with locally irregular decompositions. We are interested in both optimisation results and algorithmic problems. We first introduce an equitable version of the neighbour-sum- distinguishing edge-weightings, that is a variant where we require every edge weight to be used the same number of times up to a difference of 1. Then we explore an inject- ive variant where each edge is assigned a different weight, which yields necessarily an equitable weighting. This gives us first general upper bounds on the equitable version. Moreover, the injective variant is also a local version of the well-known antimagic la- belling. After that we explore how neighbour-sum-distinguishing weightings behave if we require sums of neighbouring vertices to differ by at least 2. Namely, we present results on the smallest maximal weight needed to construct such weightings for some classes of graphs, and study some algorithmic aspects of this problem. Due to the links between neighbour-sum-distinguishing edge weightings and locally irregular decompositions, we also explore the locally irregular index of subcubic graphs, along with other variants of the locally irregular decomposition problem. Finally, we present a more general work to- ward a general theory unifying nsd edge-weightings and locally irregular decompositions. We also present a 2-player game version of neighbour-sum-distinguishing edge-weightings and exhibit sufficient conditions for each player to win the game.