If there's one generally accepted rule of warfare, it's that you should never invade Russia during the winter. Hitler tried it and failed horribly, Napoleon tried before that and found equally terrible results, and the Swedes who fought in the Great Northern War would tell a similar story.
Supply lines running thin in the freezing cold and enveloping mud spells doom for anyone attacking into a Russian winter — or does it? For some reason, history tends to overlook the many times Russia has lost in the cold, despite their home-turf advantage.
1. The Japanese — Russo-Japanese War
Because the giant nation's borders have changed throughout history, it's hard to pinpoint what exactly constitutes the "Russia" part of a "Russian winter." Most historians would define it as invading west of the Steppes, but technically, the Japanese attacked Russia by taking Russian-controlled Korea and Manchuria.
Japan invaded and conquered the Korean peninsula in February 1904. Ironically, Tsar Nicholas II couldn't get the supplies needed from the Western half of Russia due to intense winter weather — the same conditions that, supposedly, make Russia impregnable. As a result, the Japanese were able to fortify and held the territory until the end of WWII.
2. The Finns — Continuation War
As hard fought as the Winter War between Finland and the USSR was, the Finns managed to hold onto their independence by ceding 11% of their bordering lands to the Soviet Union. Later, Finland sought to regain these lands by making an enemy-of-my-enemy pact with Nazi Germany in 1941.
Finnish forces pushed through to Leningrad so "successfully" that it made Hitler confident he could do the same. Except, in this case, "success" meant that cannibalism wasn't too widespread. Though trying, the Finns were able to hold onto territory until 1944, when Finland sided with their archenemy, Russia, to fight off Nazi Germany.
3. The Swedes — Ingrian War
Swedish invasions combined with ongoing Polish aggression (detailed below) in the early 17th century kicked off what has since been known in Russian history as "the Time of Trouble." Sweden sought to capture the Russian throne, and they started by launching an offensive on Novgorod, which resulted in the successful installation of a Swedish monarch.
To find eventual peace, treaties were formed and broken and reformed and rebroken and finally reformed in favor of Sweden. Either way, the Swedish Kingdom pushed the Russians back to Kola and, in the process, kicked off what the Swedes call their "Age of Greatness." Eventually, the Russians and Swedes formed an uneasy alliance because both of them were more focused on another common enemy: Poland.
4. The Poles — Livonian War
Not to be outdone by the Swedes, Poland also got involved with conflict in the Russians around the turn of the 17th century. Swedish and Polish armies invaded Russia from different fronts and would eventually fight each other. The Polish-Lithuanian Commonwealth get most of the credit for invading Russia because they seized and held territory in the name of Roman Catholicism.
The Polish king, Stefan Bathory, lead a widely-successful, five-year campaign against Ivan IV (or, as history knows him, Ivan the Terrible). It would take years for Russian forces to reunify under the Romanov dynasty and defend the Kremlin.
5. The Central Powers — WWI
During WWI, the Germans managed to push the Eastern front all the way to Petrograd (St. Petersburg) and caused enough instability to forever change Russian history. Germany pressured Russia into the Treaty of Brest-Litovsk in March 1918, which made Russia give up control of Poland and the Baltic States after fighting through the long winter of 1917-1918, effectively putting an end to fighting on the Eastern Front.
This was also what would sway public support for the Communist Red Army. In a way, the Czardom of Russia was completely destroyed because they lost a war fought in a Russian winter.
6. The Mongols — The Golden Horde conquests
And, of course, the grandsons of Ghengis Khan very successfully curb-stomped the Kievan Rus' at the height of the Mongol Empire. They cleared out Ryazan and Suzdal in December 1237 and eventually pushed their way into Kiev by December 1240. They were without supply lines (they were nomads — they didn't rely on them) and were very much on Russian soil as winter set in. Their primary means of battle, the cavalry, were very susceptible to the rigors of winter, but still dominated.
Nearly every harsh element of harsh winters should have crippled the Mongol forces. Instead, the Mongols only stopped because they had little interest in holding territory.