In the past few decades the fractional Laplacian (−∆)s has capitalized the attention of the PDE community as an integrodifferential operator which is the nonlocal counterpart to the classical Laplacian. Gaining popularity thanks to the seminal work of L. Caffarelli & L. Sil- vestre “An extension problem related to the fractional Laplacian”, it sees a widespread use thanks to its ability to model phenomena in which long term interactions between objects occur, leading to applications in particle physics, finance and population dynamics among others. An interesting line of research is to investigate what happens to the opera- tor as the index s approaches 0. It is in this instance that the logarithmic Laplacian L∆ pops up, as the first order term in the Taylor expansion of the fractional Laplacian as s goes to 0. The goal of this seminar is to present this fairly new operator, showcasing its basic features together with some open questions. Lastly, I will present symmetry results for an overdetermined and a rigidity problem involving the logarithmic Laplacian recently obtained in collaboration with N. Soave.