This is sort of an add-on question from my other post: https://www.reddit.com/r/AskHistorians/comments/3vcr1y/what_went_wrong_with_russiaussr/
When the USSR officially fell in 1991, how much of it became the Russian Federation? I'm talking about government structure, foreign policy, military/intelligence agencies.
I just read George Kennan's "Long Telegram" and it struck me that a lot of what is said in there seems to pertain to modern Russia. Things like aggressive foreign policy, heavy control of all media, and especially what is said on page six of this link.
It seems that the main things that did change was a somewhat more open society, along with dropping the whole communism thing.
In terms of government institutions, the Russian Federation inherited a great deal from the USSR. I'm going to talk primarily about the security and intelligence apparatus here, as that's my main area of interest. I'll leave the question of broader institutional and policy continuity to others.
As the dissolution of the USSR reached its endgame in 1992, most of the all-union (USSR-wide) institutions were abolished, but in many cases they were almost immediately re-established in new forms: regional portions of the Soviet military and security establishments formed the kernels of state institutions in the new post-Soviet republics; the intelligence agency of Belarus, for example, is still called the KGB today. For obvious reasons, the lion's share of Soviet institutions were inherited by the Russian Federation, which is generally regarded as the legal successor state to the USSR.
The involvement of the KGB — most notably the chairman, Vladimir Kryuchkov — in the attempted coup d'etat in August 1991 sealed that agency's fate. Boris Yeltsin moved to abolish the KGB as a unified service, handing over its many responsibilities to multiple new agencies of much more limited scope:
The dismantling of the KGB was a significant step, both functionally and symbolically, in the fall of the USSR, because it represented a comprehensive tearing-down of the single most powerful organ of Soviet state control; an ignominious end for the service which had long seen itself as 'the sword and shield of the Party'. It was both a repudiation of Soviet-era repression and a stratagem designed to neutralise a key part of the old guard's power base.
Conversely, the Ministry of Internal Affairs (MVD), which had long played an important role in maintaining the USSR's internal security, survived the fall of the Soviet Union essentially intact. The MVD was responsible for both the militsiya (the regular civil police), the Internal Troops (a paramilitary gendarmerie) and several special police units (eg. OMON). Interior minister Boris Pugo and a significant number of rebel MVD troops were actively involved in the August coup; Pugo committed suicide in order to avoid arrest when the putsch failed. His deputy minister, Andrei Dunaev, meanwhile, had refused to join the coup attempt, instead backing Yeltsin and bringing with him a large portion of the militsiya in the Moscow region.
Another key component of the Soviet intelligence and security apparatus survived the dissolution intact, however: the military intelligence service, GRU, by far the Soviet Union's largest (and arguably its most effective) intelligence agency, exists to this day.
All that being said, the Putin era has seen something of a reversal; Soviet-era secret service men are still to be found in powerful positions in the Russian state, and the FSB has become a powerful coercive force. Andrei Soldatov and Irina Borogan's book The New Nobility does a good job of tracing the legacy of the Soviet security state and its tendrils in the post-dissolution Russian Federation.
A couple of other good sources are Amy Knight's Spies Without Cloaks: The KGB's Successors and Louise Shelley's Policing Soviet Society: The Evolution of State Control. Masha Gessen's biography of Putin, The Man Without a Face also gives some useful context on Putin's relationship to the security services and their place in his regime.