Louis Pasteur published Germ Theory in 1861, observing that "microbes" present in the air made liquids rot. He proved microbes could be killed by heat (pasteurisation)
Robert Koch identified that different microbes caused different diseases. First discovered cholera in 1883. He came up with methods to study bacteria (grow in petri dish, stain with dye to make easier to see)
In Britain, doctors led by Henry Bastian did not believe in Germ Theory. They still believed in Spontaneous Generation (microbes spread from rotting matter by miasma). The British government rejected germ theory until the end of the 19th century