OSNAZ (ОСНАЗ) is an acronym for [voiska] osobovo naznacheniya ([войска] особого назначения) or "special purpose [detachments]", and originally were special forces troops within the KGB (its predecessors and its successor, Federal Security Service) and the MVD.

OSNAZ has always been shrouded in a veil of mystery and remains so even to this day. There is no OSNAZ headquarters as the various groups were formed within the various directorates as needed.

During the Great Patriotic War the OSNAZ of the NKVD fought on the frontlines against the Nazi Germany as VDV in the form of Airborne regiments/Brigades, with some units achieving "Guards" status at the end of the conflict.

KGB/FSB OSNAZ
The known KGB OSNAZ units were as as follows:

Alpha Group or Spetsgruppa A - formed in 1974 in the Seventh Chief Directorate
Beta Group or Spetsgruppa B- also formed in the Seventh Chief Directorate, time unknown
Vympel or Pennant - formed in August 1981 within the First Chief Directorate as a "sabotage" and covert action unit behing enemy lines. Mission redefined in the late 90s.
Kaskad or Cascade - formed in the 8th Directorate, time unknown
Grom or Thunder - VIP/VVIP Protection Unit
Delfin or Dolphin - a combat diver "incursion" unit similar to the U.S. Navy SEALs.
It is known that several have survived the fall of the Soviet Government and still exist, with the Alpha Group being the most famous or infamous, depending on point of view.

MVD OSNAZ
The Ministry for Internal Affairs (MVD) has several OSNAZ units, like SOBR, Vityaz and "Rus". They act under the aegis of their corresponding GUVD MVD (Main Internal Affairs Directorate of the Interior Ministry). They are not secret or classified units, even if their operations may be classified.

Caveat: OMON is not an OSNAZ unit. Regular police forces (militsiya) also have their special purpose units (OMSN). MVD OSNAZ units and MVD Interior operative troops (such as ODON operatives) usually wear green or olive berets, the most skilled servicemen being awarded with a purple beret (краповый берет, krapovy beret) after succeeding in a tough competition held every year. The latter OSNAZ members are commonly called krapoviki.

SVR OSNAZ
Zaslon is a possible name for a suspected and highly secret OSNAZ organization within the Foreign Intelligence Service (Russia).

FSB (Russia)
The FSB (ФСБ) is a state security organisation in Russia, and is the domestic successor organization to the KGB. Its name is an acronym from the Russian Federalnaya Sluzhba Bezopasnosti Rossiyskoy Federatsii (Федера?льная слу?жба безопа?сности Росси?йской Федера?ции) which translates to Federal Security Service of the Russian Federation. It is usually simply called the FSB in English-language sources. Its headquarters are located in Moscow.

MVD
The Ministerstvo Vnutrennikh Del (MVD) (Министерство внутренних дел) was the Ministry of Internal Affairs in the imperial Russia, late USSR, and still bears the same name in the Russian Federation. The Ministry is headquartered in Moscow.
The Russian MVD was recreated as the MVD of the Russian SFSR in 1990, following the restoration of the republican Council of Ministers and Supreme Soviet, and remained when Russia gained independence from the Soviet Union. It currently controls the Militsiya, the road police (GAI), and the internal troops. Since the dismission of the Tax Police, it also investigates economic crimes.

The long-time additional duties of the Imperial MVD and NKVD, such as the Firefighting Service and Prisons Service, were recently moved to the Ministry of Extraordinary Situations and the Ministry of Justice respectively. The last reorganization abolished Main Directorates inherited from the NKVD in favor of Departments.

SVR Foreign Intelligence Service (Russia)
SVR stands for Sluzhba Vneshney Razvedki (Служба внешней разведки), and is Russian for, "Foreign Intelligence Service". It is Russia's primary external intelligence agency. The SVR is the Russian intelligence agency, which evolved from the KGB after the collapse of the Soviet Union.

The CSR, for Centralnaya Sluzhbza Razvedki (Central Intelligence Service) took over the intelligence gathering and analysis duties of the KGB's First Chief Directorate in October of 1991.

In December of that year, the former chief of the KGB First Chief Directorate, Yevgeni Primakov, was appointed head of the organisation, which was renamed SVR.

The SVR performs offensive intelligence gathering operations abroad and also enters into formal cooperation and intelligence-sharing arrangements with foreign intelligence agencies.

The SVR are suspected of assassinating Zelimkhan Yandarbiyev, former leader of Chechnya.

GRU is the English transliteration of the Russian acronym ГРУ, which stands for "Гла?вное Разве?дывательное Управле?ние" (Glavnoe Razvedyvatel'noe Upravlenie), meaning Main Intelligence Directorate. The full name is GRU GSh (GRU GenShtaba, i.e. "GRU of the General Staff". )The GRU was created in 1918 by Lenin, and given the task of handling all military intelligence. It operated residencies all over the world, along with the SIGINT (signal intelligence) station, in Lourdes, Cuba, and throughout the former Soviet bloc countries, especially in Lithuania, Latvia, and Estonia.

The GRU was totally independent of most other power centers in the Soviet Union, most famously the CPSU and KGB. At the time of the GRU's creation, Lenin ordered the Cheka (predecessor of the KGB) not to interfere with the GRU's operations. The rivalry between the GRU and KGB was even more intense than the rivalry between the FBI and CIA.

The existence of the GRU was not publicized during the Soviet era. It became widely known in Russia, and the West outside the narrow confines of the intelligence community, during perestroika, in part thanks to the writings of "Viktor Suvorov" (Vladimir Rezun), a GRU agent who defected to Britain in 1978, and wrote about his experiences in the Soviet military and intelligence services. According to Suvorov, even the Communist Party general secretary couldn't enter GRU headquarters without going through a security screening.

The GRU still remains to this day a very important part of the Russian Federation's intelligence services, especially since it was never split up like the KGB was(The KGB was actually drastically sized down).

Federal Protective Service (Russia)
In the Russian Federation, the Federal Protective Service was formerly the Ninth Chief (aka The Guards) Directorate of the KGB and is now an independent organization. It is responsible for the protection of Russian state property and high-ranking government personnel, including bodyguard services. It also operates the secure high-level communications system and the secure subway system used by the government. It may still contain over 30,000 uniformed personnel plus several thousand plainclothes personnel. The Presidential Security Service or PSS is responsible for protecting the President of Russia and is occasionally merged with and separated from the FPS.

Russian Federation
The Russian PSS is the agency charged with the protection of the President of Russia and the Kremlin. It may be decended directly from the Kremlin Guard Force of the KGB. It is directly responsible for and to the President of Russia, it has at times been merged with and separated from the Russian Federal Protective Service or FPS. Like the United States Secret Service, the PSS contains both uniformed and plainclothes personnel. Like the FPS service in the PSS is equal to military service.

OMON or Otryad Militsii Osobogo Naznacheniya (Russian: Отряд милиции особого назначения, ОМОН; literally: Special Purpose Detachment of Militsiya) is a generic name for the system of special units of militsiya within the Russian and earlier the Soviet, Ministerstvo Vnutrennih Del (MVD; Ministry of Internal Affairs). Commonly called the Black Berets (even if this term seems to be reserved to the Russian naval infantry), there is an OMON unit in every subnational entity of the Russian Federation. The units are utilized in emergencies such as high-risk arrests, hostage crises and riots, as well as in response to acts of terrorism. OMON are often accused of rash actions and excessive and indiscriminate use of force.

Militsiya (Russian: мили?ция; Ukrainian: мiлiцiя; Romanian: Militia; literally "militia") was the generic name for the police in the Soviet Union and a few other Communist countries. It is now used as a short official name of the police in Russia, Ukraine, Belarus and some other post-Soviet states. Due to the history of the term and the distinctive local features, the militsiya should be considered a special regional kind of policing system, not just a translation of the English "police". Militsiya forces in all post-Soviet countries share similar traditions, tactics and methods, although the differences are increasing over time.