Hi. I live in south Florida and I had a child with my boyfriend. Unfortunately, things didn't work out and we broke up. Now, he's threatening me to take my child away. His name shows on my daughter's birth certificate. What legal rights does he have?

The same legal rights as any father. If he can show that you are an unfit mother, he could gain custody. But that's a big IF. Courts are reluctant to remove children from the mother unless the child is in danger in the mother's care.

Many father's will threaten and bluster about pursuing primary custody to intimidate the mother into not trying for child support. So don't fall for that. The more likely situation is he will get some level of visitation and you will get support. So file for support.

He can file for custody, but unless mother unfit, normally will not get.
Although it is possible.

He can file for joint custody and this is becoming more and more popular
And he can file for visits

Same as any father