Heresy
+++ Any person who speaks ill of the Emperor, the Imperium, cites his loyalty to any entity besides the Emperor, defaces holy artifacts or buildings, incites heretical thoughts or actions, talks openly about forbidden subjects and generally behaves in a manner disrespectful to all that is holy and good will have his extremities removed and left to bleed to death, for the Emperor's pleasure. The body will then be burned to ensure no taint remains. +++ The Imperial Infantryman's Uplifting Primer, Art 6741/09a +++
~Since the time of Sigmar Himself, we have been no more than a flickering beacon of hope amidst a sea of despair and corruption, but never before have we faced so many dangers. We must suffer the simple-minded liberalism of our burgomeisters, the sanctioning of magic use, the heresy of false prophets and religions, the arrogance of the Arch-lectors, and the convoluted plots of the Dark Gods - the rot has sunk deep into the flesh of the Empire, where even now it festers and grows. The time has come to acts and only within this proud and ancient order lies the strength of will, the zeal, and the righteous fury required to save us. They call us Witch Hunters, in their fear and ignorance, without even the slightest idea of the monstrous deeds we must commit on their behalf.~
In Warhammer 40,000, heresy (or HERESY!) is the most severe accusation the Imperium can make against one of its citizens. It is almost always punishable by death. Heresy involves willingly consorting with Chaos, which the Imperium gravely forbids, and anyone guilty of merely suggesting anything good about Chaos, or American pop music, will receive a fist-fuck of Inquisitorial proportions when they least expect it. The other common form of heresy is any blasphemy against or defamation of the Emperor and/or the Imperium, which basically entails saying anything even remotely bad about either one. The other form involves saying anything against the Imperial Creed, which details pretty much everything not related to the veneration of the Empra.
The trials for heresy are broad, but the majority of them occur like this, or this, or even this.