Elastic and diffractive scattering in the LHC era

We review elastic and diffractive scattering of protons (called also "forward physics") with emphasis on the LHC data, especially those deviating from the expectations based on extrapolations from earlier measurements at the ISR, Fermilab and thus triggering searches for new ideas, models and theories. We list these new data and provide a brief introduction of available theoretical approaches, mainly those based on analyticity, crossing symmetry and unitarity, particularly the Regge-pole model realizing these concepts. Fits to the data are presented and tensions between theoretical predictions and the data that may indicate the way to further progress are in the focus of our paper.