All the ways? I will give it a shot.:) 1) "to want"("~하고 싶다") 잠자+고 싶다 : I want to sleep. 2) "A and B"("A 하고 B") 내 친구 하+고 나는 저녁을 같이 먹을꺼야: My friend and I are going to have dinner together. In here "~하고" is the same as "~랑" or "~와/과". 3) "After~"("~하고 나서") 저녁을 먹+고 나서 서점에 갔다 : After I had dinner,I went to a bookstore. 4) "~and~"("~하고 나서 그리고") 저녁을 먹+고 나서 서점에 갔다 : I had dinner and went to a bookstore. 5) "~ with somebody" ("somebody 하고 같은/이") "너 하+고 같은 생각이야" : I agree with you. "너 하+고 같이 갈께" : I am going with you. 6) When you introduce somebody or yourself.("~라고 하다.") 저는 파스칼이라+고 합니다 : This is Pascal or My name is Pascal. It is used when you refer(or address) somebody or yourself to others. 7) In a reported speech ("~ 다/라고 하다") (그는) 오늘저녁에 못온다+고 했습니다.: He said he won't be able to make it tonight. Just 6 usages came across to my mind.:) I think "~고" by itself is meaningless, you should memorize it with some phrases like "~하고 나서" or "~라고 하다" or "somebody 하고 같은~" etc.. one more, 8) something is happening ("~하고 있다") 자+고 있다 : "He is sleeping" 먹+고 있다 : "to be eating" 공부하+고 있다 : "to be studying"

To add to Pascal's answer, I learned a from a helpful italki member that 이러고 + 고 있다 or 이러고 + __(verbstem) + 고 있다 is sarcastic. The of equivalent of a when we make a comment in English and raise our brows for censure, then saying " You're doing that/being like that...? or 이러고 놀고 있다 " You are playing around like that...?" here's the link to my journal post this, he explains it much better lol. http://www.italki.com/entry/224648