This talk focuses on the reconstruction of unknown coefficients in the wave equation, leveraging the power of Carleman inequalities. We aim at designing a global reconstruction algorithm for coefficients – such as time-independent potentials or wave propagation speeds – from Neumann boundary measurements of the solution. We begin by revisiting the historical foundations of uniqueness and stability results for this inverse problem, established over 25 years ago using local and global Carleman estimates. Building on these insights, we present a more recently developed reconstruction algorithm that exploits the same technical tools to ensure global convergence, avoiding possible local minima encountered by conventional methods. The algorithm is grounded in the minimization of a functional derived from the Carleman weight function, directly inspired by the stability proof strategy. The talk will outline the key steps in proving the algorithm’s convergence for reconstructing coefficients in bounded domains or networks. These works are the result of various long-standing collaborations with Maya de Buhan, Emmanuelle Crépeau, Sylvain Ervedoza, Axel Osses, and Julie Valein.