We use can, could, be able to, and may with an infinitive (e.g. work, dance, sing):

• Aldona can dance the Tango now.
• Last year she couldn’t dance at all.
• She’ll be able to dance in the school competition soon.

When we talk about a person’s ability to do something, we normally use can (negative cannot or can’t) for present time, could for past time, and will be able to for future time.

In past time, we do not normally use could for something that happened on a particular occasion. We use was able to or managed:

• The plane was in difficulties, but in the end, it managed to land at the nearest airport. (or … it was able to land …; not … it could land …)
• We were able to get a really good price on the car. (or … managed to get …; not … could get …)

We use Can I etc. to ask for permission to do something. Note that Could I and May I are more formal and polite than Can I.

We can use can or may to give permission.

• You can open the window. (or You may open the window.)

If we talk about what is allowed in general (i.e., not by a particular person), we use can:

• People can drink alcohol when they are 18.
• You can smoke in this area.

But official notices often use may:

• CARS MAY BE PARKED HERE FOR A MAXIMUM PERIOD OF ONE HOUR.

We use Can you, Could you, and Would you (but NOT May you) when we ask somebody to do something. Could you and Would you are more formal and polite than Can you.

• I’ll take out the rubbish.
• Shall I open the window?

We use Shall I and I’ll to offer to do something. Note that Shall I is a question.

• A: Shall I turn on the light?
• B: Yes, please. / No, thank you.