Fusion fast ignition (FI) initiated by a laser-driven particle beam promises a path to high-yield and high-gain for inertial fusion energy. FI can readily leverage the proven capability of inertial confinement fusion (ICF) drivers, such as the National Ignition Facility, to assemble DT fusion fuel at the relevant high densities. FI provides a truly alternate route to ignition, independent of the difficulties with achieving the ignition hot spot in conventional ICF. FI by laser-driven ion beams provides attractive alternatives that sidestep the present difficulties with laser-driven electron-beam FI, while leveraging the extensive recent progress in generating ion beams with high-power density on existing laser facilities. Whichever the ion species, the ignition requirements are similar: delivering a power density ≈1022 W cm−3 (∼10 kJ in ≈20 ps within a volume of linear dimension ≈20 µm), to the DT fuel compressed to ∼400 g cm−3 with areal density ∼2 g cm−2. High-current, laser-driven beams of many ion species are promising candidates to deliver such high-power densities. The reason is that high energy, high-power laser drivers can deliver high-power fluxes that can efficiently make ion beams that are born neutralized in ∼fs–ps timescales, making them immune to the charge and current limits of conventional beams. In summary, we find that there are many possible paths to success with FI based on laser-driven ion beams. Although many ion species could be used for ignition, we concentrate here on either protons or C ions, which are technologically convenient species. We review the work to date on FI design studies with those species. We also review the tremendous recent progress in discovering, characterizing and developing many ion-acceleration mechanisms relevant to FI. We also summarize key recent technological advances and methods underwriting that progress. Based on the design studies and on the increased understanding of the physics of laser-driven ion acceleration, we provide laser and ion-generation laser-target design points based on several distinct ion-acceleration mechanisms.