The activity of Task 5.2 “Pushing Accelerator Frontiers” (PAF) supports the community effort in pushing the hadron-beam brightness. Bright hadron beams are fundamental for atomic and nuclear physics experiments as well as for achieving novel ultra-precise nuclear clocks (HITHOR). Multi-species beams in storage rings are foreseen as a novel intriguing possibility for collisions in a moving frame, and new insights into nuclear reactions. Space-charge forces and intra-beam scattering are two fundamental mechanisms that limit the ultimate beam brightness. Pushing Accelerator Frontiers has co-organized the “Space Charge 2024” workshop where the state of the art in this domain has been reviewed.