The Earth's inner core translational oscillation modes， also referred to as Slichter modes， are the basic normal modes of Earth's free oscillation， and the periods of the Slichter triplet are important physical quantities which can be used to determine the density jump across the inner core boundary， and the latter is of great research value to constraining the density structure of the deep interior of the Earth. However， there are not generally accepted conclusions so far about the Slichter modes' excitation mechanism and attenuation mechanism as well as the actual detecting results， which lead to great arguments. Hence， the determination of the eigenperiods has become one of the international challenges in fundamental geophysics. This paper firstly summarizes the basic theories of the Earth's inner core translational oscillation modes， including their dynamic equations and main solving theories and numerical methods， and overviews their theoretical periods under different Earth models using different solving theories. Secondly， the main hypotheses and conjectures about Slichter excitation mechanism and attenuation mechanism are discussed， and thereinto， degree-one surficial pressure flow acting in the core may excite the Slichter modes to an observable level. Finally， we review the research progress in the study of the detections of the Slichter triplet signals using the superconducting gravity data in the past 30 years， and discuss some potential future research subjects about the Slichter triplet detection. Therefore， from three different perspectives including exploration of excitation mechanism， fine preprocessing of superconducting gravity data， stacking and enhancement of extremely weak signals， the research breakthrough is expected to achieve reliable detection of the Slichter triplet signals.