It is Russian tradition that woman takes surname of her husband (Russian idiom "взять фамилию", "взять фамилию мужа"). But she also can keep her maiden (original) surname. Also, sometimes (rare, but possible) man takes surname of his wife (usually when she has beautiful surname). Also possible have double surname. For example, if Ivan Ivanov takes to wife Daria Petrova, she could take double surname and become Daria Ivanova-Petrova. For man is the same option. But this cases are rare, mainly woman takes husband's surname or keeps her maiden surname. Also, after divorce woman can restore her maiden surname if she wishes. It is not common, but may be used as symbol "I don't want to have anything common with him, even surname"

Many women do change their surnames after marriage, but not always. If woman likes her surname she might not want to change it and it's okay. If wife has the same surname as her husband (for example, she is Maria Ivanova and he is Ivan Ivanov - in that case she would not change her surname because it is the same).

Yes I know that the father's name will be the patronym and so on... Maybe an example?