He has no wife and he has no children. This a key thematic point near the end of the story. He has noone to whom he can pass his kingdom. He dies alone. He has no physical legacy - part of the reason he wants to fight the dragon himself. He wants to leave the treasure and fame as part of his legacy.
Since he has no children, Wiglaf is named king.

There are some indications that Beowulf married Queen Hygd after she becomes a widow, but it is not well fleshed out in the text. I'm not sure about kids.