The urgency of the problem of ensuring asteroid impact avoidance today is obvious. Modern observational tools make it possible to detect and investigate dangerous objects, and the development of space systems allows solve the issue of preventing their collisions with The Earth. Such events as a meteorite fall near Chelyabinsk in 2013 make it possible to understand the importance of the problem not only to specialists, but to all the inhabitants of our planet. Ensuring asteroid impact avoidance is a complex, interdisciplinary problem. It includes observations of small bodies, determination of their orbits, allocation of dangerous bodies, determination of the characteristics of possible collisions with The Earth. Astronomers of all over the world are working on it. We note the great contribution of American experts working under the auspices of NASA. In addition to this astronomical part, it is necessary to highlight measures for minimization of damage in the event of an asteroid collision with The Earth. In Russia the Ministry of Emergency Situations (EMERCOM) deals with it. Finally, collision avoidance measures - the most complex and costly and least developed today - focusing on changing of an asteroid orbit (or asteroid destruction). Basically, these activities are developed theoretically. We also note the joint NASA and European Space Agency mission "AIDA", which would target Didymos, a binary asteroid system, with a 300-kg spacecraft and study the corresponding change in the orbit. The fact that the asteroid is binary allows us to conduct these observations more accurately. Currently, a huge number of scientific articles and monographs are devoted to the problem of asteroid hazard; conferences that are fully or partially devoted to it are held regularly as well. We indicate several recently published works in Russia 1], 2], 3]. This paper discusses some features of the movement of hazardous asteroids, important for understanding the evolution of their trajectories and the possibilities of preventing collisions with The Earth.