permission, requests, suggestions, offers, invitations 	

Asking permission
Could I have a look at that?
May I know the reason?
Can I borrow the car tomorrow night?

Asking  someone to do something 
Can you give me a hand with this table?
Would you like to get the conference room ready?
Could you take these up to Fanny's office?
Would you make some tea?
Would you mind taking these down to reception?
Will you let me know as soon as possible?

Inviting someone to do something
Will you join us for a drink afterwards?
Won't you stay to dinner?
Would you like to go for a walk this afternoon?

Making a suggestion
We could change the colour, I suppose.
Shall we take a break?
We could always postpone the meeting until next week?
We could change the colour, I suppose.
Shall we get started?

Offering to do something
Would you like me to call them?
Shall I ask him for you?
I'll get this – it's my treat.
Can I give you a hand with that?

Offering something
Would you like some more cake?
Have another one, go on!
Will you have another drink?