Thesis: This thesis studies the centralized assignment of teachers to schools and a new matching framework inspired by it. In the first chapter, we develop a theoretical model of reassignment to study the problem of reassigning tenured teachers who already have a position and are willing to move to another school. The problem is similar to the one of assigning students to schools. In this case, the well known Deferred Acceptance algorithm has been identified as the only algorithm that: i) is stable ii) efficient and iii) gives incentives to students to report their true preferences. The main difference with the problem of assigning students to schools is that teachers have an initial assignment. One has to consider an additional constraint, Individual Rationality (IR): a teacher must receive a school that he weakly prefers to his initial one. To incorporate this constraint, a modification of the Deferred Acceptance algorithm has been identified in the academic literature and used in practice to assign teachers to schools in France. We show that this modified algorithm has a serious drawback: it is not efficient in a strong sense. Indeed, it is possible to reassign teachers to schools such that both: i) teachers obtain a school that they prefer and ii) schools are assigned teachers that they rank higher. Thus, we identify the class of all algorithms, the Block-Exchange (BE) algorithms, that do not suffer from this drawback. Among them, we show that there is a unique one that gives good incentives to teachers to report their true preferences, the Teacher Optimal Block-Exchange algorithm (TO-BE). In using a large market setting, we theoretically show that these algorithms perform better in terms of movement and welfare for teachers than the currently used one. We then use a dataset on the assignment of teachers to schools in France in 2013 to quantify the possible gains that can bring our algorithms. In a reassignment setting with no newly tenured teachers or empty seats, we show that we can more than double the number of teachers obtaining a new assignment. In the second chapter, we aim to design a practical algorithm, inspired by our findings in the previous chapter, for the French assignment system of teachers to schools. More generally, this design also aims to provide a tool about two important issues common to OECD countries: i) the lack of attractiveness of the teaching profession and ii) the high achievement inequality between students from different social backgrounds. We consider the complete French market composed of tenured teachers looking for a reassignment, newly tenured teachers with no initial assignment and empty positions. In improving the mobility of teachers, one can give them better career perspectives and so potentially attract more teachers into the profession. But in doing so, it can also hurt deprived regions in assigning more teachers with low experience to them and ultimately the students from these regions. We propose a flexible algorithm that allows to better control the movement and distribution of teachers across regions, especially deprived ones. Using the data of the French assignment of teachers in 2013, we simulate several counter factuals and show that our algorithm can accommodate a wide range of policy objectives.