"The goal of this ERC Starting Grant proposal is to make significant advances in our understanding of how planetesimals and gas giant planets form. I propose an ambitious research programme dedicated to answering three key questions at the frontier of planet formation theory: - How do mm-sized particles grow past the bouncing barrier? - What is the Initial Mass Function of planetesimals? - How do the cores of gas giants form and evolve?} I will address these questions using a combination of novel ideas and computer simulations to model three critical stages of planet formation: 1) the growth of pebbles into rocks and boulders by coagulation and vapour condensation, 2) the gravitational collapse of clumps of rocks and boulders into planetesimals with an array of sizes, and 3) the long term growth of planetesimals as they grow to become cores of gas giants by accreting pebbles embedded in the gas. These investigations will form an important theoretical foundation for understanding the next generation of observations of protoplanetary disc pebbles, planetesimal belts, and planetary systems. The self-consistent models for the formation of planets resulting from this proposal will shed light on the spatial distribution of pebbles in gas discs around young stars (observable with the ALMA telescopes), on the initial state of planetesimal belts (crucial for understanding the evolution of debris discs observable with JWST and the asteroid and Kuiper belts), and on the formation and evolution of the wealth of exoplanetary systems detected in the near future (by astrometry with the Gaia satellite, by ground-based radial velocity surveys, and by direct imaging with E-ELT)."