Is turing completeness related to recursive enumerable languages?

Question

I recently came across this term "Turing complete" in my study of morphogenesis models. When I looked up, it said that a turing complete machine can simulate a universal turing machine.

When I looked up for its relationship with recursive enumerable languages, I couldn't find anything substantial.

My question is:

Will a turing computable machine always accept recursive enumerable languages?

Thanks in advance!