What were the Nuremberg Trials?

There were several Nuremberg trials. The most well known Nuremberg trial was the one of the significant Nazi leaders such as Goering, Speer, von Papen and the like. This was an Allied trial where Russian, British, French and American judges presided. Whilst the trial attempted to be fair and unbiased, the trial of the defeated by the victors was always going to cause some difficulties, and to this day the validity and legality of the trials is questioned. Issues such as the lack of a right to appeal, and the lack of any trial for Allied officers, leaders and soldiers who committed acts that were considered war crimes when committed by Nazis meant that the trial was often viewed as unfair and simply a way of 'rubbing salt into the wound'. Regardless of this, some of the accused were found innocent and freed, although many of them were simply arrested by the new German government and imprisoned anyway!

However, there were subsequent Nuremberg trials, such as the trial of German judges that was depicted in a black and white film a few decades ago, that were wholly American trials. They also tried other professionals such as Nazi doctors. The total list of the Subsequent Nuremberg Proceedings, as they were known, follows.

Trial 1: The Medical Case: Doctors experimenting on prisoners, sterilization of prisoners etc.
Trial 2: The Milch Case: Only one defendant, named Milch, regarding the Government-sponsored slave economy.
Trial 3: The Justice Case: Judges in applying the racial laws (also enacted in Nuremberg under the Nazis - this was one of the reasons the trials were held in Nuremberg).
Trial 4: The Pohl Case: Oswald Pohl, headed the section of the government that ensured that all hair, gold teeth etc. was removed from concentration camp victims, and also helped administer the 'death by overwork' policy. He was executed in 1951.
Trial 5: The Flick Case: Friedrich Flick, wealthy Nazi supporter, used slave labor to make money, was released in 1951, refused to compensate surviving slaves and died a billionaire several decades after the war.
Trial 6: The I.G.Farben Case: I.G. Farben were the producers of Zyklon B, the main gas used at the concentration camps. They also used slave labor in their factories. I.G.Farben executives and managers were tried and imprisoned.
Trial 7: The Hostage Case:
Trial 8: The RuSHA Case: The Race and Resettlement Office (RuSHA), oversaw the forced movement of large groups of people into concentration camps and other countries. This was a trial of its more senior members, I believe, but I have been unable to find very much information about this case.
Trial 9: The Einsatzgruppen Case: Dealt with senior members and leaders of the Einsatzgruppen, or death squads, who were active in Russia. These often ignored groups actually killed more Jews than many concentration camps - it is believed that the death squads killed over a million Jews, although the accurate records found at concentration camps were not maintained by these mobile death squads. They were also the first groups to begin experimenting with gassing, trying several different types of gas in mobile gas vans prior to the creation of the gas chambers at other concentration camps.
Trial 10: The Krupp Case: I only know that this was regarding the use of slave labour by private industry.
Trial 11: The Ministries Case: This was regarding the many different laws and policies enacted by the government, and the turning of the population against the Jewish people through government sponsored and created literature.
Trial 12: The High Command Case: Like the name suggests, this was a case for high ranking officers, such as those who directed the actions of the Einsatzgruppen from Germany.

So, the trials were both Allied, as well as American!