Second-order Gauge-invariant Cosmological Perturbation Theory: Current Status updated in 2019

The current status of the recent developments of the second-order gauge-invariant cosmological perturbation theory is reviewed. To show the essence of this perturbation theory, we concentrate only on the universe filled with a single scalar field. Through this review, we point out the problems which should be clarified for the further theoretical sophistication of this perturbation theory. This review is an extension of the review paper [K.~Nakamura, ``Second-Order Gauge-Invariant Cosmological Perturbation Theory: Current Status'', Advances in Astronomy, vol.2010 (2010), 576273.]. We also expect that this theoretical sophistication will be also useful to discuss the future developments in cosmology as a precise science.