1. Ability
1.1. Present
1.1.1. Can
1.1.1.1. Affirmation/Negation
1.1.1.1.1. I can climb that tree
1.1.1.1.2. I can't eat shellfishes
1.1.1.2. Question
1.1.1.2.1. Can I take a pencil from your case?
1.2. Past
1.2.1. Could
1.2.1.1. Affirmation/Negation
1.2.1.1.1. Finally, I could finish the project
1.2.1.1.2. I couldn't finish the project
1.3. Future
1.3.1. (Will) be able to
1.3.1.1. In 2030, I will be able to speak german
1.3.1.1.1. II
1.4. Conditional
1.4.1. Could
2. Advice
2.1. Should
2.1.1. Suggestion
2.1.1.1. You don't look well; you should go to the doctor
3. Possibility
3.1. May (Colloquial)
3.1.1. She may go to the supermarket tomorrow
3.2. Might (Formal)
3.2.1. He might sign the contract
4. Obligation
4.1. Must
4.1.1. Can't refuse
4.1.1.1. You must do your homework
4.2. Have to
4.2.1. Softer. Necessary, but not an obligation
4.2.1.1. For the project, you'll have to bring a cardboard
4.3. Need to
4.3.1. Even softer. A personal need
4.3.1.1. I need to pass that exam
5. Prohibition
5.1. Mustn't
5.1.1. Not allowed
5.1.1.1. You mustn't smoke under the age of 18