Recent advances in nanotechnology have made a whole zoo of particles of different shapes available for applications, but their interaction with biological cells and their toxicity is often not well understood. Experiments have shown that particle uptake by cells is determined by an intricate interplay between physicochemical particle properties like shape, size, and surface functionalization, but also by membrane properties and particle orientation. Our work provides systematic understanding, based on a mechanical description, for membrane wrapping of nanoparticles, viruses, and bacterial forms. For rod-like particles, we find stable endocytotic states with small and high wrapping fraction; an increased aspect ratio is unfavorable for complete wrapping. For high aspect ratios and round tips, the particles enter via a submarine mode, side-first with their long edge parallel to the membrane: For small aspect ratios and flat tips, the particles enter tip-first via a rocket mode.