How do you say "proud of you" in Chinese? As in if a child does something good, their parents might say "I'm so proud of you" or "Mum is so proud of you".

The only word in the dictionary I seem to be able to find for pride seems to be about self-pride （eg. 自豪).

Google Translate translate pride as 骄傲 but I always thought that 骄傲 kind of had negative connotations like of someone being a bit arrogant or vain.
骄傲 is completely correct, you can use 骄傲 in this situation, it won't bring any negative connotations.
In some situations, 骄傲 really means arrogant, for example, &quot;This guy is so 骄傲&quot;, &quot;这个人很骄傲&quot;.
Below statements fulfill &quot;Mum is so proud of you&quot;:

妈妈为你感到骄傲
妈妈为你感到自豪
妈妈为你感到光荣

To be more emphasize, you could use &quot;无比&quot;，it is like incomparable, for example:

妈妈为你感到无比骄傲
妈妈为你感到无比自豪
妈妈为你感到光荣无比