Which should I say:
I'm looking forward to our meeting.
I look forward to our meeting.
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These are both correct. I would add, though, that I'm looking forward to our meeting sounds (to me, at least) more conversational (and a bit more genuine), whereas I look forward to our meeting is a bit more formal/polite. I would expect to find the I'm looking form in spoken language, and the I look form in writing (likely at the end of an email confirming a meeting).