a) Ability: "can", "could", "be able to"

a.1. "can" / "could"

1.- We use "can" ("could" en el pasado) to say that something is possible or to say that someone knows how or has the ability to do something.

can (could) + infinitive

I can see the lake from my window

Can you speak Japanese?

We can't go to Paris this weekend

"Can" can only be used in the Present and the Past. It cannot be used in the present perfect. Can does not have an infinitive.

a.2. "be able to"

2. We can use "be able to" instead of can.

be able to + infinitive

I haven't been able to do my homework. I will do it tomorrow

Are you able to play the piano?

b) Obligation: "must", "have to", "have got to"

b.1. "must"

must + infinitive

1.- We use "must" to say that it is necessary for someone to do something

I must go to work today

They mustn't smoke in this building

We use "must" when we feel that something is true

You have been working all day. You must be tired

Manuel has been translating all day. He must be bored

For the past tense we use:

must + have + past participle

I have lost my mobile phone. I must have left it in the library

b.2. "have to + infinitive" / "have got to + infinitive"

1. We use "have to" / "have got to" when you talk about external obligation, such as when a law or someone says that something is necessary or important.

Tonight we have to go to swimming practice (if we don't we won't be in the team)

They have got to go to school on Friday

c) Possibility

c.1. "may" / "might"

may (might) + bare infinitive

They may go to the party on Saturday

I might wear a dress

We may not be able to go to school this week

She might not (mightn't) play tennis at Wimbledon

1.- We use "may" or "might" to talk about possible actions in the future.

2. "May" and "might" are usually interchangeable. However, we use "might" (not "may") when the action is not real.

If I knew your Mum better, I might invite her over for a cup of tea (The situation is not real because I don't know the Mum better, so I'm not going to invite her. May is not possible in this example).

d) Responsibility

should (ought to) + bare infinitive

e) Lack of obligation / responsibility

don't have to (needn't) + bare infinitive