The free translational oscillations of the inner core, the so-called Slichter modes have been a subject of observational controversy. Its detection has never been undoubtedly validated. Also, it motivated additional theoretical studies. The search for the Slichter modes was invigorated by the development of worldwide data recorded by superconducting gravimeters (SGs) of the Global Geodynamics Project. Thanks to their long-time stability and low noise level, these relative gravimeters are the most suitable instruments to detect the small gravity signals that would be expected from the Slichter modes. The theory is now better understood and the most recent computations predict eigenperiods between 4 h and 6 h for the seismological reference PREM Earth model. A more recent study states that the period could be much shorter because of the kinetics of phase transformations at the inner-core boundary (ICB). The observation of the Slichter modes is fundamental because, the restoring force being Archimedean, their periods are directly related to the density jump at the ICB. This parameter is still poorly known. The analysis from seismic PKiKP/PcP phases or from normal modes observation leads to discrepancies in ICB density contrast estimates. This parameter should satisfy both the constraints set by powering the geodynamo with a reasonable heat flux from the core, and PKP travel-times and normal mode frequencies. This paper gives a review of the theoretical backgrounds as well as of the attempts to detect such free oscillations. Some possible excitation sources are also investigated to evaluate the expected amplitude of the Slichter modes. The seismic source has been previously studied to demonstrate that an earthquake of magnitude Mw = 9.68 would be necessary to excite the Slichter mode to the nanogal level (≈ 10-12 g where g is the mean surface gravity) at the Earth's surface. Earthquakes are therefore not the most suitable source to excite the Slichter mode to a level sufficient for the SGs to detect the induced surface gravity effect. Surficial pressure flows acting in the core have also been considered as a possible excitation source. The later turns out to be the best way to excite the translational motion of the inner core. However we have little information about the fluid pressure amplitudes acting in the core at those frequencies. The observation of this Earth's normal mode is still an open challenge and the development of new generations of instruments with lower noise at such frequencies would be the only chance of detection.