Why did the Soviet Union invade Hungary in 1956 and Czechoslovakia in 1968?

No country west of USSR were Soviet allies. They were occupied, and enslaved countries with enforced totalitarian structure that was against the wishes of the majority of the people. These nations never accepted Soviet control of their country by hostile power as USSR as their own. USSR conquered this part of the world with the blessing of the Western power and controlled them until 1989, when it colonial empire collapsed.

Hungary was a country that was historically against Russians and was part of Nazi coalition during WWII. It was defeated in 1944 by Red Army and occupied as state that waged a war against USSR. USSR treated Hungary as occupied enemy and continued to loot this country as war repayment till the Hungarian uprising in 1956. Soviet Union maintained military presence in Hungary after WWII and used to suppress the uprising. Hungarian revolution was directed against USSR and against government that was a representative of USSR in Hungary. Soviets could not allow Hungary to go Yugoslav way.

Czechoslovakia was a different scenario than Hungary. USSR did not really cared about internal politics of the country nor for its liberal course that started WELL BEFORE 1968. Czechoslovakia did not had Soviet troops in the country as Red Army withdrew in 1945 together with American Army which was stationed in western Bohemia. After communist coup in 1948, Czechoslovakia maintained close relationship with USSR, but was otherwise isolated totalitarian country. People in general never really accepted communist takeover, but tolerated as long the living standard was high (and it was higher there till 1960's than in Austria). In the 60's the communist party lost control over the country, especially in media, culture, art, civic organization, and in 1968 also in politics. The same time nationalism between Czechs and Slovaks flared up. Brezhnev words about the Czechoslovak crisis "It is your matter", encouraged anti-soviet dissidence. In summer of the 1968, there was a call for Czechoslovakia to withdrew from a Warsaw Pact and create own military alliance with Yugoslavia and Romania as was before WWII. This step was proposed in the week of the Soviet invasion, and USSR stroke first. Letting Czechoslovakia to leave Warsaw pact would render it crippled and together with Romania it would undermine Soviet power west of Moscow. USSR wanted to prevent collapse of its imperial domain and took over Czechoslovakia by military force. And with Czech territory bordering Bavaria, Soviet could instal military bases that were capable to shoot anywhere into Western Germany instead from distant Poland, Hungary or East Germany.

Poland was never communist country, but people's republic where even communist party did not accepted Soviet control. The chief of the military in Poland was Russian Rokossovski, because Soviets distrusted Polish Army. Poles had multiple times showed lack of loyalty to USSR (1956-1970-1980), and Poland was only country in the Eastern Block that was capable to wage a war with USSR if the situation would be dire. Soviets waging a war in Afghanistan were unable to invade Poland, plus with Reagan, the war would go bad for Warsaw Pact. Polish resistance to Soviet rule was too strong, and USSR pressured Polish military to suppress the resistance, which had happened after the coup December 13, 1981.

Russia Invades Czechoslovakia

Simply because they wanted to opt out of Communism and become independent of Soviet Russia and Russia, who viewed their 'empire' as the precursor to World Domination and World-Wide Communism, wouldn't stand for a moment letting any country succeed from the Soviet Conquest.

The Soviets Truly believed that their ideology of Communism was a formula which should be implemented all over the world, replacing free Enterprise and Capitalism. Unfortunately, by no stretch of the imagination could Soviet Russia actually be defined as 'communist.' It was not..! It was pure 'Stalin-ism,' and the Cult of The Dictator. True Communism never had a chance in Russia. Stalin and his henchmen saw to that.

The Soviet Union acted as it did in order to suppress movements in those countries that could have led to democratic governments that would have broken free of being Soviet satellites.

October 1956 - a spontaneous uprising of the hungarian people against oppression. It was suppressed with Russian troops.

Why did the Soviet Union invade Hungary in 1956 and Czechoslovakia in 1968?

and threaten to invade Poland in 1980, if those countries were supposedly its allies?

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1. Hungary in 1956.....Hungary was an ally of Hitler's Nazi Germany and invaded Soviet Union in 1941 togeather with Hitler - so in 1945 it was occupied the same as all other Hitler's alies - USA and UK occupied West Germany, West Austria and Italy, while USSR occupied East Germany, East Austria, Hungary, Romania and Slovakia (all those were alies of Hitler and took part in agression against USSR)....So in 1956 locals in Hungary rebelled against occupation and Soviet Union helped those commies who sided with Soviet to subdue the uprising - actually Soviets didn't take part in fighting, they just helped local pro_Soviet fraction with armor...The contraversy was that althoug Hungary was still a territory of occupied Nazi Axis power, Soviets put up with Hungarians trying to make them as friends and allies, thats why it was like double standards situation -were they occupied and defeated enemy and then Soviets were right to crush revolt...or were they normal and equal people who wanted to get rid of "paternalism" of equal to them country? - as there was just 10 years paste their war against Soviet Union and Allies, majority of rebells were indeed former soldiers who invaded USSR, so....I would not blame Soviets in this case in Hungary

2. Czeckoslovakia 1968...that was totally different situation ....though Slovakia was an agressor against USSR in 1941 and was defeated and occupied (like Hungary by Soviets or Italy and Austria by Brits and Americans ), but Czechia was a victim of German agression and so was considered as liberated by Soviets,not occupied - there were no Soviet troops in Czeckoslovakia (even from Slovakia those were withdrown in 1948)...But as Czeckoslovakia was a Warsaw Pact member, USSR was afraid it would join NATO if liberals would topple Communist Giovernment there - so in agreement with Czeckoslovakian COMMUNIST GOVERNMENT the Warsaw Pact (not only Soviets, but Hungarians, Romanians ,Poles, Bulgarians and east-Germans too) mooved their troops to Prague....there were a lot of demonstartions and protests, but no blood was spilled...In contrast to Hungary, actions in Czeckoslovakia were illegal (even though negotiated with Czeckoslovakian Government, but ofcource there was pressure and Czecks agreed under threats...)

3. Poland.....well, there were aproximatelly the same situation as in Czeckoslovakia, but Soviets never did anything - Polish communists and government themselves subdued Solidarity moovement for a while...for 9 years at least....

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gee bee wrote : "Simply because they wanted to opt out of Communism and become independent of Soviet Russia and Russia, who viewed their 'empire' as the precursor to World Domination and World-Wide Communism"

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Russia was the same way NOTHING in Soviet Union as any other republic - Ukraine or Georgia or Belarus etc...and even with less self-government then vassal states of Poland, Hungary etc... - Soviet Russia was a vassal province of USSR(Soviet Union) and even names of Russian Prime Ministers and presidents in times of USSR are not known in the west as they had no any say in politics of Soviet Union (exept Yeltsin, but only because he managed to get National Independance for Russia in his fight with Soviet President Gorbachev) - so don't bs. about Russia here - it was Soviet Union and its Intenational by content Politburo that consisted of commies of all ethnicities that ruled all over the places, not was it Russia - and even the head of it (of Soviet Union) who started the whole mess in East Europe - Stalin, was not a Russian, but a Georgian - as is well known.

To crush anti Communist independence movements in these countries.