Worries about this happening were part of the Red Scare movement during the 1950s. For Cold War versions, this will invariably be a World War III scenario. If they succeed, this will result in the world becoming a Commie Land. For the post-Cold War period, the writers will have to Make the Bear Angry Again.
Subtrope of Take Over the World. Compare America Takes Over the World, China Takes Over the World and Japan Takes Over the World. Unlike those examples, Russia almost always uses its military to take over the world instead of via cultural or economic hegemony.
- In the Elseworlds story Superman: Red Son, the Soviet Union expands into a global empire under Superman's Big-Brother style leadership, who creates a totalitarian "utopia" to protect humanity from itself. After Superman's "death" it is replaced by Lex Luthor's Global United States.
- The 2000 AD story Invasion! explores this idea. Written in the 1980s at the end of the cold war, it explored the idea of a Russian invasion and takeover of Western Europe and Britain. (The Russians were thinly disguised as the "Volgans".) This was expanded into the later graphic novel series Savage.
- Exploited but ultimately subverted in the Ender's Shadow series. Achilles, looking to Take Over the World, sees Russia returning to militarism and allies himself with them promised to make them a great empire. While Russia does increase in power under his command, including forming a New Warsaw Pact, Achilles has his eye on bigger fish, and sells them out to trade himself over to India, and then to China. In the end he's defeated and a global union is formed instead to command the world.
- In the Dante Valentine novels the Russian-derived Putchkin Alliance is one of the setting's two Space Filling Empires, controlling most of Asia. (The apparently American-derived Hegemony controls the rest.)
- Nineteen Eighty-Four: One of the world's three superpowers is Eurasia, the Russian-derived successor state that controls the former Soviet Union and continental Europe (but not the British Isles). They are allied with Oceania against Eastasia until Oceania betrays them by teaming up with Eastasia against Eurasia. Apparently practices "Neo-Bolshevism" and according to the propaganda of Oceania is currently leading a campaign to conquer all of Africa. This is a subversion, since neither Eurasia nor any of the other two superstates really desire world domination, only to control their own peoples.
- In the future history presented in Cities in Flight, Russia made peace with the (by then just as oppressive) USA in the early 21st century. When the American president was assassinated, the Russian premier declared himself ruler of both countries. The resulting empire grew to become a world government, known as the Bureaucratic State.
- Amerika is a miniseries set some years after the Soviets have taken over the United States and installed a puppet government. References to other parts of the world indicate the Soviets are firmly in charge elsewhere as well, with the Warsaw Pact stretching all of Europe and China ceding Manchuria to the USSR.
- Sliders: The pilot episode involves an alternate timeline where the Soviet Union took over most of the world, including the United States.
- Command & Conquer: Red Alert Series: The first game is set in an Alternate Timeline where Einstein uses time travel to prevent Hitler's rise to power, leading to a Soviet conquest of Europe. Later games see the Soviets attempt to take their conquest global.
- Empire Earth: The game's Russian campaign is set 20 Minutes into the Future where the paramilitary group Novaya Russia seizes power and begins a war of expansion against their neighbors. Helped by robotic troops and the US's isolationism, they're able to conquer Asia, Europe, and Africa, and start moving on Cuba when a disillusioned soldier defects to the US and travels back in time to prevent the fascist regime from ever rising, only to find that future soldiers already got there and need to be eliminated with inferior modern troops.
- Rise of Nations: The expansion campaign Thrones and Patriots has a Cold War campaign where you can diverge heavily from history to ensure that the Soviets dominate the globe.
- American Dad!: In "The Best Christmas Story Never", Stan goes back in time believing that killing Jane Fonda will prevent the liberalization of the holiday season. When Stan came back to the present, he found that his efforts caused Ronald Reagan to lose the presidential election to Walter Mondale, who just "gave" America to the Soviets.
- Zigzagging Trope in real life. While the Russian Empire at its height did cover about 1/6 of the Earth's landmass, no Russian state has seriously entertained the notion of taking over even Europe, let alone the world. Russian geopolitics is often characterized as 'defensive imperialism' because it is driven by a perceived vulnerability to neighbouring states rather than simple desire for conquest. The Eurasian plains are a wide open area with few natural barriers, so Russian leaders have considered it necessary to directly control the numerous access points utilized by invading armies.
Some of the October revolutionaries did envision a global Soviet Union, but primarily effected through a worldwide revolution of the working classes, not through Russian conquest alone. For a long time Lenin even looked to Germany as the inevitable leader of the worker's revolution (it was, after all, the birthplace of Karl Marx, as well as perceived as the most industrialized western nation) until he realized that it wasn't gonna happen, and Stalin eventually codified the doctrine of "Communism in one country". As it is an anti-nationalist ideology at its roots, it's extremely doubtful that the theorized communist world state would privilege any particular nation.
If you have an EXTREMELY forgiving definition of "empire" however and saw all Communist states in the Cold War as an extension of Moscow's will, then the Russians came closest to this in the Cold War with numerous allied and satellite states. That is until the Sino Soviet split which also led to greater independence for Yugoslavia and Albania. However, the Soviet Union at the height of its influence was led by Stalin, a Georgian (albeit a Russian loving Georgian).