Every year near-Earth object (NEO) surveys discover hundreds of new asteroids, including the potentially hazardous asteroids (PHA). The possibility of impact with the Earth is one of the main motivations to track and study these objects.

This paper presents a tether assisted methodology to deflect a PHA by connecting a smaller asteroid, altering the center of mass of the system, and consequently, moving the PHA to a safer orbit. Some of the advantages of this method are that it does not result in fragmentation, which could lead to another problem, and also the flexibility to change the configuration of the system to optimize the deflection according to the warning time.

The dynamics of the PHA-tether-asteroid system is analyzed, and the amount of orbit change is determined for several initial conditions. Only motion in the plane of the orbit of the PHA around the Sun is considered, thus the PHA chosen for the simulations has low orbit inclination.

Analysis of the dynamics of the system shows that the method is feasible for planetary defense.

(Via Dynamics of tethered asteroid systems to support planetary defense)

Science fiction writers have been thinking about moving asteroids for generations. In his playful, tongue-firmly-in-cheek 1958nstory And Then the Town Took Off, Richard Wilson describes the creation of a new planet - from the asteroids:

Magnology is safe, stressless, and permanently powerful in stasis. It is the ultimate in gravity-beam nullification. If anything can glue the asteroids back into the planet they once were, magnology will do it.
Read more about building a planet with asteroids)

Robert Heinlein's 1939 story Misfit featured the movement of a selected asteroid to an ideal location for a base. Using a ballistic calculator

"Now about our job -- We didn't get one of the easy repair-and-recondition jobs on the Moon, with week-ends at Luna City, and all the comforts of home. Nor did we draw a high gravity planet where a man can eat a full meal and expect to keep it down. Instead we've got to go out to Asteroid HS-5388 and turn it into Space Station E-M3. She has no atmosphere at all, and only about two per cent Earth-surface gravity..."

[Asteroid] Eighty-eight swung some millions of miles further around the sun. The pock-marks on her face grew deeper, and were lined with durite, that strange close-packed laboratory product which (usually) would confine even atomic disintegration. Then Eighty-eight received a series of gentle pats, always on the side headed along her course. In a few weeks' time the rocket blasts had their effect and Eighty-eight was plunging in an orbit toward the sun.
(Read more about moving an asteroid)