For engineering, in general, there is civil, computing, chemical, biomedical, mechanical and electrical engineering (this is definitely not the full list, just the ones I can think of now).
Even within biomedical engineering there are a few different fields. You already mentioned genetic engineering. There is also tissue engineering, medical device development, …
Engineering is amazingly diverse! Theres so many (mechanical, electrical, chemical, aeronautical for example) and then they all split up more but theres also ones which are mixes of everything else, its really cool.