The linear complexity of a de Bruijn sequence is the degree of the shortest linear recursion which generates the sequence. It is well known that the complexity of a binary de Bruijn sequence of length 2 n n n is bounded below by 2 n0 n0 n0 1 + n and above by 2 n0 n0 n0 1 for n n n 3. We briefly survey the known knowledge in this area. Some new results are also presented, in particular, it is shown that for each interval of length 2 blog nc blog nc blog nc + + + 1 , in the above range, there exist binary de Bruijn sequences of length 2 n n n with linear complexity in the interval.