Historically, has it always (or usually) been the case that children of slaves were born into slavery themselves? For example, were the children of Roman slaves also slaves or were they considered free persons? I was just wondering if this was relatively unique to American slavery.
The children of slaves are the property of the slave owner and has always been since there has been chattle slavery.
You should have looked it up yourself.
Roman Slaves
Thanks for the reference, even though you felt compelled to preface it with a snide remark.

Since you seem to have missed its salient point, I will repeat it for those who have a genuine interest in this topic:
If a slave married and had children, the children would automatically become slaves.
I don't know how accurate this statement is, but I will accept it at face value. However, there did seem to be some avenues for emancipation which were not available to American slaves. (I am just trying to gain an historical perspective on this subject.)