1. Galileo believed earth revolved around sun in conflict what the Church believed - sun revolved around earth, as a result Galileo was put under house arrest
More on Galileo http://en.wikipedia.org/wiki/Galileo_Galilei
2. Charles Darwin theory of Evolution that all live evolved, humans evolve from apes, this is against the Church’s view that God created the world and humans in ‘His image and likeness’ ( Book of Genesis)