Short Japanese Culture Questions

Short question about family names.
How does it work with getting married and having kids?

Let’s say that 岡崎 道子 (OKAZAKI Michiko) falls in love with 青木 猛 (AOKI Takeshi).
They get married. Does Michiko becomes 青木 道子? And when they get kids, do they also get 青木 as a last name?