Transition probability density function (TPDF) or log-TPDF of a diffusion is quite useful in many ways. For example, it can be employed not only to estimate a diffusion by the maximum likelihood estimation but also to simulate data from a diffusion or to price an asset when the underlying process follows a diffusion. However, unfortunately, the true TPDF of a diffusion is unknown with a few exceptions in general. Starting from Aït-Sahalia (2002)'s pioneering work on approximate but explicit TPDF of a univariate time-homogeneous diffusion to Choi (2019a)'s recent work on closed-form approximate TPDF of a multivariate time-inhomogeneous jump diffusion, several researchers have subsequently established the way to approximate the TPDFs or log-TPDFs of more general diffusion models. This article explains how people have resolved problems to generalize the method from Aït-Sahalia(2002)'s paper to Choi(2013, 2015)'s multivariate time-inhomogeneous diffusions. Due to space constraints, explanations of detailed theories or assumptions for their proof are reduced to the minimum and we show important results, with tacit facts not described in the original papers. In addition, we also introduce papers derived from and related to those key studies.