I present a novel framework for carrying out global analyses of the Standard Model Effective Field Theory (SMEFT): SMEFiT. This approach is based on the Monte Carlo replica method for deriving a faithful estimate of the experimental and theoretical uncertainties and enables one to construct the probability distribution in the space of the SMEFT degrees of freedom. As a proof of concept of the SMEFiT methodology, we present a first study of the constraints on the SMEFT provided by top quark production measurements from the LHC. I then discuss an complementary approach to constrain the SMEFT parameter space space based on Bayesian inference. I also describe the first steps towards a joint determination of the SMEFT degrees of freedom and of the proton structure, an essential ingredient for the interpretation of future LHC precision measurements. Finally, I outline the next steps in a program aiming to construct a first global analysis of the SMEFT.