Methods and apparatus to grasp an object with an unmanned aerial vehicle are described herein. An example unmanned aerial vehicle includes a gripper having a claw to grasp onto an object and an active material disposed on the claw. The example unmanned aerial vehicle further includes a material activator to: (1) apply an activation signal to the active material to soften the active material while the claw grasps the object with the active material, and (2) allow the active material to harden in a shape substantially matching a surface of the object.