According to Abraham Ascher in his book Russia: A Short History, Stalin had ample warning but chose not to believe the information he was getting.

Winston Churchill (and other people) warned Stalin that Germany intended to invade the Soviet Union, but Stalin didn't believe that Hitler would betray him and violate the Non-Aggression Pact. A deserter from the German army told Russian border guards about Operation Barbarossa on June 21, 1941, but "Stalin dismissed the report as an obvious provocation and ordered his commanders to be on the alert but not to take any action" (Ascher 196). He had heard that Hitler planned to invade, but simply didn't believe it~~--he reportedly went into a state of shock for about a week after the Germans launched their offensive~~.

As for why Stalin didn't think that Hitler would betray him, I don't know and Ascher unfortunately doesn't go into more detail, but I hope this answers your question!

EDIT: A poster below told me that the week-long state of shock that Stalin reportedly went into is an urban legend not corroborated by reliable sources from the time.

They noticed. But what did Hitler intended to do? We know that he intended to invade, pretty much whatever Stalin did. Historically that's not the way it usually works. So many people thought that Hitler would issue an ultimatum. Stalin wanted to delay war with Germany. If he could comply with the ultimatum without compromising the Soviet Union's military strength, he would agree and Hitler would not invade. If he couldn't agree to the ultimatum it would at least serve as a warning.

My source here is Barton Whaley, Codeword Barbarossa, MIT press, 1974

This is my first time actually answering a question in this subreddit, so if I break the rules somehow, or I am just flat out wrong, please remove the post mods.

Actually, quite the contrary. They did know of the massive military preparations along their borders. According to Antony Beevor in Stalingrad:

"In the Soviety Embassy on the the Unter den Linden. . . An urgent signal from Moscow demanded 'an important clarification' of the huge military preparations along the frontiers from the baltic to the Black Sea."

Furthermore, the Kremlin at this point in time (21 June, 1941) was actually freaking out about the entire thing, and had started to realize just what the intentions of the German military were. However, Stalin did not believe that their intentions were to invade the USSR, he merely believed that Hitler wanted to "extract greater concessions" from him. Stalin also seemed to believe that the idea of Hitler invading was simply a campaign started by the British.

"Stalin was certain that most warnings had been. . . part of a ploy by Winston Churchill, the arch-enemy of the Soviet Union, to start a war between Russia and Germany. Since [Rudolph] Hess's flight to Scotland, the conspiracy had grown even more complicated in his mind."

And, lastly, it seems that Stalin had actually been petrified of provoking Hitler in anyway, and refused to even accept that an invasion was possible. According to Beevor,

"Stalin, the totalitarian dictator, still could not come to terms with the idea that events [Operation Barbarossa] might be outside of his control."

So, in short, Stalin was surprised by operation Barbarossa because he had refused to believe that it was even a possibility. In his mind, he believed that Hitler was just trying to gain concessions from him, that the British were behind the entire rumor that the USSR would be invaded, and he was afraid of Hitler and did not want to provoke him.

I hope that this answers your question!

Source:

Beevor, Antony. Stalingrad. London: Viking, 2007. 3-6. Print.

I just got done reading William Shirer's omnibus, The Rise and Fall of the Third Reich (Simon & Shuster, 1960), and he notes that not only did Stalin know of the troop build-up from some of the sources already cited here, but that he was also explicitly warned by U.S. Undersecretary of State Sumner Welles. Further proof that he knew of the troop build-up is laid out by Shirer through eye-witness testimony from Nuremburg that related how Molotov bluntly asked the Nazis about their troop movements in the East.

As to why Stalin ignored them? Still no great answer for you; he must not have read Mein Kampf?