In this review paper on heavy ion inertial fusion (HIF), the state-of-the-art scientific results are presented and discussed on the HIF physics, including physics of the heavy ion beam (HIB) transport in a fusion reactor, the HIBs-ion illumination on a direct-drive fuel target, the fuel target physics, the uniformity of the HIF target implosion, the smoothing mechanisms of the target implosion non-uniformity and the robust target implosion. The HIB has remarkable preferable features to release the fusion energy in inertial fusion: in particle accelerators HIBs are generated with a high driver efficiency of ~30%-40%, and the HIB ions deposit their energy inside of materials. Therefore, a requirement for the fusion target energy gain is relatively low, that would be ~50-70 to operate a HIF fusion reactor with the standard energy output of 1 GWof electricity. The HIF reactor operation frequency would be ~10-15 Hz or so. Several-MJ HIBs illuminate a fusion fuel target, and the fuel target is imploded to about a thousand times of the solid density. Then the DT fuel is ignited and burned. The HIB ion deposition range is defined by the HIB ions stopping length, which would be ~1 mm or so depending on the material. Therefore, a relatively large density-scale length appears in the fuel target material. One of the critical issues in inertial fusion would be a spherically uniform target compression, which would be degraded by a non-uniform implosion. The implosion non-uniformity would be introduced by the Rayleigh-Taylor (R-T) instability, and the large densitygradient- scale length helps to reduce the R-T growth rate. On the other hand, the large scale length of the HIB ions stopping range suggests that the temperature at the energy deposition layer in a HIF target does not reach a very-high temperature: normally about 300 eV or so is realized in the energy absorption region, and that a direct-drive target would be appropriate in HIF. In addition, the HIB accelerators are operated repetitively and stably. The precise control of the HIB axis manipulation is also realized in the HIF accelerator, and the HIB wobbling motion may give another tool to smooth the HIB illumination non-uniformity. The key issues in HIF physics are also discussed and presented in the paper.