PoS(CORFU2021)106 Running Vacuum and the ΛCDM tensions Joan Solà Peracaula * Departament de Física Quàntica i Astrofísica, and Institute of Cosmos Sciences, Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain E-mail: sola@fqa.ub.edu In the last few years a lot of work has provided significant support to the possibility that the vacuum energy density (VED), ρ vac , is a running quantity throughout the cosmological history. Recent theoretical studies have shown that the properly renormalized ρ vac in FLRW spacetime adopts the ‘running vacuum model’ (RVM) form, in which the scaling with the renormalization point turns into dependence on the Hubble rate, H. The late time VED evolves as an additive term plus a dynamical component O (H 2 ). Higher (even) powers O (H 2n ) are also predicted, which can trigger inflation in the early universe, although we shall not discuss this part here. In addition, the VED running is free from the quartic powers of the masses of the fields (∼ m 4 ) and hence the cosmic evolution of ρ vac is really smooth. On the phenomenological side, the RVM fits the cosmological data remarkably well and it may help to reduce the H 0 and σ 8 tensions afflicting the ΛCDM. Overall, the RVM is sound since its theoretical structure can be derived from quantum field theory in curved spacetime and the model is phenomenologically consistent. Corfu Summer Institute 2021 "Workshop on the Standard Model and Beyond," (CORFU2021) 29 August - 8 September 2021 Corfù, Greece * Speaker. c  Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/