The article examines the evolution of nuclear deterrence and strategic stability concepts, including their role in security policy of the United States and the USSR (Russia). The works of B. Brodie, T. Schelling, A. Wohlstetter, H. Kahn are the subject of analysis from the point of their contribution to both: the theory and the practice of implementation of this policy at different historical stages of security relations between the two nuclear superpowers. Special attention is payed to the contradictory role of ballistic missile defense in respect of strategic stability theory. The author comes to the conclusion that strategic stability always had as minimum two ‘visions’. The first one (the author calls it ‘MAD Stability’) completely rejects any positive role which can be played by ballistic missile defense in promoting more stable security environment. The second one (‘BMD Stability’), to the contrary, suggests that limited BMD systems can only strengthen security and deterrence since it plays very positive role in minimizing the risk of unauthorized and accidental use of nuclear weapons. Alongside with it BMD can complicate the task to deliver a first ‘disarming’ strike, makes less possible limited nuclear options and can protect against the threat from the third states as well as against terrorist attacks. The author shows that during all the period of nuclear age there was a struggle between a ‘common sense approach’ to BMD and the approach, based at ‘paradoxical’ vision of strategy. The first one was expressed by Soviet Prime Minister A. Kosygin in 1967, who rejected the idea to put a ban on the deployment of antiballistic missile defense since it is designed ‘to protect human lives’. The same idea was behind President’s R. Reagan ‘Strategic Defense Initiative’ of 1983. But the major period of time the Parties officially supported a ‘MAD Stability’ approach, the symbol of which was the ABM Treaty. After the US withdrew from this Treaty in 2002, Russia continued to follow this ‘interpretation’ of strategic stability which brought a lot of suspicions and critiques of the US ‘plans to achieve military superiority’. The author comes to a conclusion that after Russia established in 2011 Air and Space Forces with BMD tasks, the Parties could find common grounds for better understanding and possible future agreements on security issues.