The muon anomalous magnetic moment, $a_{\mu} = (g_{\mu}-2)/2$, was measured with a precision of 0.54~ppm
at Brookhaven National Laboratory (BNL) around 15 years ago. The measured value differs from the Standard Model (SM) prediction by more than three standard deviations. This may imply existence of new physics beyond SM. The new J-PARC experiment aims to measure $a_{\mu}$ and the electric dipole moment (EDM) by a completely different method to that at BNL. In the J-PARC experiment, ultracold muons are stored in a 0.66~m-diameter magnetic field without electric focusing. The ultracold muons are produced by laser-ionising muonium atoms accelerated by the muon-dedicated linear accelerator. This paper reports the current status of each experimental component.