I’ll be as short as possible, but it will still be a long post.
Today the Katyn massacre is the name for a whole series of massacres of the Polish POWs (from the POW camps Starobelsky, Kozelsky and Ostashkovsky) and prison inmates (from the prisons in the annexed parts of Ukraine and Belorussia) in the spring of 1940 committed on Stalin's order (more specifically, the 05.03.1940 order of the Politburo).
Since its ~22,000 victims included thousands of officers, including generals, and a huge chunk of the Polish elite, their disappearance (lack of correspondence) was noticed immediately, and especially after USSR failed to release them along with all the other amnestied Polish POWs. The Polish Government-in-Exile (PGE) drafted the lists of the missing persons and unsuccesfully tried to clear up their fate by gathering information from the POWs previously held in USSR as well as by asking the NKVD officials and even Stalin directly (e. g. the ± Sikorski and general Anders tried to do it). They were met with evasive answers boiling down to „we don’t know what happened to them“. Stalin hinted that some might have fled, e. g. to Manchuria. After the war some Polish officials would claim that the NKVD officials told them that some sort of a great mistake was committed vis a vis these POWs.
In early 1943 the Germans found the bodies of some oft he missing POWs in the Katyn forest near Smolensk (those were the bodies of the Kozelsky camp POWs). The worst Polish suspicions seemed tob e confirmed by the findings: the names matched the missing persons; the documents on the bodies (letters, newspapers) were dated not later than the spring of 1940; the locals confirmed that the Poles were shot by NKVD.
After the PGE called on the Red Cross to investigate the allegations and USSR immediately broke all diplomatic relations with it.
Nevertheless, the evidence was extremely problematic in the eyes oft he world. It was gathered by the Nazis and propagandized by the agencies oft he well-known liar Goebbels (indeed, the Nazis tried to blame the massacre on the Jews and even published a fake list of Jewish names of the alleged NKVD men – though it was soon forgotten). Little was known about the fact that the Polish Red Cross took part in the exhumations and in ist internal documentation confirmed their authenticity. Moreover, the Western Allies knew that the whole affair was only a maneuver on the Nazis‘ part to split the Allies, and thus chose to ignore the revelations.
In 1944 the Soviet investigative commission under Burdenko re-exhumed several hundred bodies in the Katyn forest and unsuprisingly established that the deed was done by the Germans. The witnesses said their previous testimonies were coerced by the Germans; several documents were allegedly found on the bodies dated long after the spring of 1940 (allegedly overlooked by the Germans). The commision’s report settled the matter for many, or at the very least reduced it to a he-said-she-said, despite the fact that it did not clear up the fate of the missing officers from the other camps.
During the International Military Tribunal in Nuremberg the Soviet Union tried to blame Katyn on the Germans, specifically it came up in the case against Goering. Internally, the Soviet judges argued that the Trial rules meant that the Burdenko report should be accepted as evidence and cannot be disputed. But the Western Judges did not let the Soviets‘ trick slide and pointed out that although the rules allow for the report to be accepted as evidence, its veracity can be disputed by the defense. The judges reached a compromise – each side would provide 3 witnesses, which then testified. It was arguably a wash, unable to convice anyone of the opposite opinion, and Katyn was not mentioned in the verdict.
During the Cold War the old political considerations did not matter anymore; in fact, Katyn could be used well against USSR. So there were several attempts to investigate it further, the most famous one being the Madden commission (The Select Committee to Conduct an Investigation and Study of the Facts, Evidence, and Circumstances of the Katyn Forest Massacre). In 1951-52 this House committee interrogated numerous witnesses and examined numerous documents. This included the reps of the PGE, the former Polish POWs (incl. Prof. Stanisław Swianiewicz who testified about seeing his fellow POWs shortly before they were brought to the Katyn forest, while he himself was taken off the transport), the people who took part in exhumations, and even the former American Nuremberg prosecutor Jackson.
Albeit not without flaws (e. g. the commission examined an anonymous witness whom we know to be fake today – he testified about having witnessed the shooting; they also took some of the false rumors about the Ostashkov POWs having been drowned in barges somewhere seriously), the general picture painted by the commission was credible – there was the problem of the missing POWs and the only plausible answer was that the Soviet Union did it. The Soviet Union largely ignored the event, mostly reissung the Burdenko report as the answer.
Various researchers took up the topic from time to time, but there was not much new to add to the results of the Madden commission. A fake document, the so-called Tartakov report, seemingly confirming the shooting by the NKVD (most probably forged by the Polish emigre Mieczysław Gorączko), surfaced in the 1950s and was used by various researches (like Louis FitzGibbon), but that was that.
The real breakthrough came in the late 1980s, when the Soviet historians Lebedeva, Parsadanova et al. found and analyzed the archive of the Main Administration for Affairs of Prisoners of War and Internees in the Special Archive in Moscow. They noted that the traces of the POWs end in the spring of 1940 with their deportations somewhere, e. g. in case of Kozelsk they were sent to the train station Gnezdovo (near Katyn). The sequence of the layers in the Katyn graves could also apparently be somewhat correlated to the transportation lists. The Burdenko report was re-examined and its inconsistencies pointed out (e. g. no POW camps with designations „ON“ existed). It became clear that the POWs really disappeared from the internal Soviet records after the spring of 1940, which was extremely telling. The "missing Poles" issue, relevant already during the war, has now been confirmed by the Soviet documents. Albeit the evidence was indirect, but it was still obviously damning and had only one plausible explanation.
In 1990 Gorbachev, based on the historians‘ findings, officialy admitted the Soviet guilt. However, Gorbachev also knew about the documents in the so-called sealed envelope in the Kremlin archive and chose not to reveal them.
Meanwhile the Soviet Prosecutors‘ Office started an official investigation into the Katyn-related crimes. It turned out that Katyn was just one of several mass grave sites related to the crime. Several old ex-members of the NKVD confirmed having taken part in the massacre in various roles, the head of the Kalinin UNKVD Tokarev described the massacre in the Kalinin prison (Ostashkov POWs) in detail. Mass graves in Mednoye (at Kalinin, now Tver) and Pyatikhatki (Kharkov) were partially investigated, the results were telling – the Polish POWs were indeed found, the documents found on the bodies were dated not later than the spring of 1940.
All this has already amply confirmed the Soviet guilt, but then came a new revelation, already after the fall of the USSR, under the Russian President Yeltsin. He was given the sealed envelope by Gorbachev and decided to make it public. Among other things it contained Beria’s March 1940 proposal to shoot the Poles, with positive resolutions by Stalin and other Politburo members; the official Politburo decision; and the head of KGB Shelepin’s 1957 note proposing to destroy about 22,000 case files of the Poles shot on Stalin’s order. The documents were revealed in 1992 and officially published in January of 1993.
This brought new information to light: the victims were not only about 15,000 POWs but also about 7000 prison inmates, about which heretofore not much had been known. (Their existence was suggested by the numeration gaps in the troika protocols, and the new documents closed the gaps).
Meanwhile, the Polish prosecutorial and archeological teams made their own exhumations in Katyn, Mednoye, Pyatikhatki, Bykovnia (further confirming the Soviet guilt). And some time later the former Ukrainian KGB files in Kiev about Pyatikhatki were opened and revealed some further information about the inmates of prisons who were shot (the correspondence about the so-called Ukrainian list), and about the KGB attempts in 1969 to destroy the graves in Pyatikhatki (these internal KGB documents once again openly acknowledge the Soviet guilt).
Such is basically the state of knowledge today, very shortly.
Still not clear: the location of the graves of the inmates from UkrSSR and BSSR (except for the Bykovnia graves) and the names of the victims from BSSR (the Belorussian list is still missing).