The Kontsevich invariant is a powerful invariant of knots which dominates all quantum invariants and Vassiliev invariants of knots. It is known that the Kontsevich invariant can be expanded in the special form called the “loop expansion”. The 2 (resp. 3)-loop polynomial is the polynomial presenting the 2 (resp. 3)-loop part of the loop expansion of the Kontsevich invariant. There are some calculations and formulas about the 2-loop polynomial. On the other hand, we have few results about the 3-loop polynomial. In this talk, we review the Kontsevich invariant, the 2-loop polynomial, the 3-loop polynomial, and their properties. After that, we state the recent re sults about the 3-loop polynomial, especially, a restriction of the set of possible value of the 3-loop polynomial of genus 1 knots with trivial Alexander polynomial, and some examples of its calculation. If time permits, we see the relationship between the loop expansion of the Kontsevich invariant of knots and the LMO invariant of the cyclic branched covers of knots.