According to / in accordance with; and 'as' in the sense of 'as X happens, Y happens' — used both for following rules/instructions and to express correlated change (as A, B).
会社の規則に従って行動してください。
Please act in accordance with the company's rules.
先生の指示に従って実験を進めた。
We carried out the experiment following the teacher's instructions.
年を取るに従って体力が落ちる人が多い。
As people get older, many experience a decline in physical strength.
気温の上昇に従って氷が次第に溶けていった。
As the temperature rose, the ice gradually melted.