The phrase ‘look forward to’ causes all sorts of problems. Firstly, the phrase, as often said, is not ‘to look forward’ which means to face ahead of you and not backwards. It is ‘to look forward to’ which means to be excited about a future event. The ‘to’ is a preposition here, and is very important.
You cannot say, “I am looking forward to see you”, you have to say “I am looking forward to seeing you” – because after prepositions, we always use a gerund (-ing form). This applies to all these phrases – I am looking forward to meeting you, I am looking forward to visiting Paris etc.
And another thing – students often sign off their emails or text messages simply with the phrase, “I am looking forward”, and any native will be thinking in their head, “Looking forward to what?” Leaving the ‘to’ part off is just not allowed at all in English. If a context has already been established and both people know what you’re talking about, you have to say, “I am looking forward to it.”
A: I am looking forward to going to the party. Are you?
B: Yes, I am really looking forward to it.