Proton reconstruction with the Precision Proton Spectrometer (PPS) in Run 2 and the PPS at HL-LHC

The Precision Proton Spectrometer (PPS) started operating in 2016 and has collected more than 110 fb$^{-1}$ of data over the course of the LHC Run 2, now fully available for physics analysis. This contribution covers the key features of the PPS alignment and optics calibration, which have been developed from scratch. The reconstructed proton distributions, the performance of the PPS simulation and finally the validation of the full reconstruction chain with physics data (dilepton events) are also illustrated.