Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are closely connected to several areas in mathematics. There are a few fundamental ways of constructing all cyclic codes over finite fields, including the generator matrix approach, the generator polynomial approach, the generating idempotent approach, and the $q$-polynomial approach. Another one is a sequence approach, which has been intensively investigated in the past decade. The objective of this paper is to survey the progress in the past decade in this direction.