Which solution concepts in game theory are used in which players both compete and cooperate?

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In the Wikipedia article "Multiplayer game", it is stated in the Game theory section that

If cooperation between players is allowed, then the game becomes more complex; many concepts have been developed to analyze such games. While these have had some partial success in the fields of economics, politics and conflict, no good general theory has yet been developed$^{1}$.

I have several questions regarding this passage:

  1. Which game-theoretic concepts have been analysed in which players can both cooperate and compete with one another?
  2. In what sense has no good general theory been developed yet?
  3. Are there any game-theoretic concepts that relate to collaborative players within teams that compete against other teams? (See also this question.)

[1] This quote references the book "Moral Calculations: Game Theory, Logic, and Human Frailty".