Over the past two years noteworthy experimental and theoretical progress has been made towards the top-level scientific question for the US programme on heavy-ion-fusion-science and high energy density physics: 'How can heavy-ion beams be compressed to the high intensity required to create high energy density matter and fusion conditions?' New results in transverse and longitudinal beam compression, high-brightness transport and beam acceleration will be reported. Central to this campaign is final beam compression. With a neutralizing plasma, we demonstrated transverse beam compression by an areal factor of over 100 and longitudinal compression by a factor of > 50. We also report on the first demonstration of simultaneous transverse and longitudinal beam compression in plasma. High beam brightness is key to high intensity on target, and detailed experimental and theoretical studies on the effect of secondary electrons on beam brightness degradation are reported. A new accelerator concept for near-term low-cost target heating experiments was invented, and the predicted beam dynamics validated experimentally. We show how these scientific campaigns have created new opportunities for interesting target experiments in the warm dense matter regime. Finally, we summarize progress towards heavy-ion fusion, including the demonstration of a compact driver-size high-brightness ion injector. For all components of our high intensity campaign, the new results have been obtained via tightly coupled efforts in experiments, simulations and theory.