We develop upper and lower bounds for the numerical radius of 2 × 2 off-diagonal operator matrices, which generalize and improve on some existing ones. We also show that if A is a bounded linear operator on a complex Hilbert space, then for all r≥ 1 , w2r(A)≤14‖|A|2r+|A∗|2r‖+12min{‖ℜ(|A|r|A∗|r)‖,wr(A2)}where w(A), ‖ A‖ and ℜ(A) , respectively, stand for the numerical radius, the operator norm and the real part of A. This (for r= 1) improves on some existing well-known numerical radius inequalities.