Don't lose touch. After a breakup, you should part ways for a while, at least until the heat of the split-up subsides a bit. Putting some distance between yourself and him will allow you to think about your former relationship more clearly. If you want him back, however, you need to remain a presence in his life. Little things will suffice, like chatting online or a sending a Facebook message or email every now and then.
Go to the places you know he frequents, but make sure his current partner won't be there. Ask a mutual friend or check his Facebook page (if he has one) to find out where he will be. Put on nice clothes and look your best when you see him. Be friendly, confident and relaxed.
Call him. Ask him how he is and what he's up to. Talk about things you know he likes and that you used to share. For example, if he likes music, recommend a band you think he will like. It will give you another reason to contact him later, because you will be curious to know if the band appealed to him. Keep the conversation positive; avoid making snide comments about how he's spending his time and who he's with, and be upbeat about what's going on in your life.
Be honest. A straightforward approach is the hardest but most effective, because you will know right away what his thoughts are and if you stand any chance at all. Tell him you miss him and you want him back. Acknowledge his current relationship, and explain that while you don't want to hurt his current partner, you can't get over what you had with him.
Move on with your life. Even if you are hoping your ex-boyfriend will come back, there's no point in wallowing in self-pity while you wait for it to happen. Start a new class or hobby. Pick something that makes you fulfilled and happy; this will not only occupy your time, but will also give you confidence and open you up to new experiences, possibly making you more appealing to your ex.