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Here we summarized our recent work on evolutions of magnetic field and spin-down of neutron stars. Our work includes the following three parts: (1) Based on the estimated ages of their potentially associated supernova remnants (SNRs), we estimate the values of the mean braking indices of eight magnetars with SNRs, and find that five magnetars have smaller mean braking indices of 1 < n < 3, and we interpret them within a combination of magneto-dipole radiation and wind-aided braking. The larger mean braking indices of n > 3 for the other three magnetars are attributed to the magnetic field decay; (2) By introducing a mean rotation energy conversion coefficient, and combining the pulsar’s high-energy and timing observations with a reliable nuclear equation of state, we estimate the initial spin period, initial dipole magnetic field and true age PSR J1640−4631, The measured braking index of n = 3.15(3) for PSR J1640−4631 is attributed to its long-term dipole magnetic field decay and a low magnetic field decay rate; (3) The low braking index of n = 0.9(2) for PSR J1734−3333 could undergo a supercritical accretion soon after its formation in a supernova explosion. The buried multipole magnetic fields will merger into a dipole magnetic field. Keeping the current field-growth index ε = 1.34, this pulsar will become a magnetar with dBp ∼ 6.2(2) × 1014 G and dBp ∼ 1.06(4) × 1015 G after next 50 kyrs and 100 kys, respectively. [ABSTRACT FROM AUTHOR]