Only if he asks. There was a thread about this recently. I have looked over his inbox if he leaves it open but don't read his emails. I have read texts but not proud of it - he had a woman 'stalking' him (ok, not really, just sending weird texts and stuff, but she lives in spain so I'm not really worried...!) and I wanted to see what she had written. Very occasionally I look at his mobile inbox to check if she has text him (he wouldn't remember to tell me) but I only look at the names, not the actual messages. I'd be annoyed if he did that to me though.
However, all my passwords are saved on the computer so he could read my emails/facebook if he wanted, but I don't think he does.
I learnt a lesson about spying when I hacked into his emails a couple of years ago and misread the date of one to his ex GF, thought he was cheating on me but I read the year wrong! I felt really quite stupid