Objective: to stop or alter the trajectory of an asteroid (impact prevention) or capture it for study/resources.
-This illustration: physical  capture: wrap the object in a net/bag or attach it with harpoons (e.g. Philae/prototype harpoons, Hayabusa2 used a projectile to collect samples).
-Strike the asteroid with a spacecraft to alter its speed (e.g. DART mission). 
-Gravitational tractor: a stationary probe uses its weak gravitational pull to slowly pull the asteroid onto a new trajectory. (requires time and precise maintenance). 
- Propulsion towing: attach a cable and use engines (energy-intensive, complex). 
- Deflection by thermal flashes/ablation: focus energy (laser or mirror) to create a jet of material and push the asteroid.