Structural stability of a multiferroic compound CuFeO₂ belonging to the delafossite family has been investigated at high pressures using in-situ Raman spectroscopy. At ambient conditions, CuFeO₂ has a rhombohedral structure with space group R3m. It has two Raman active modes, identified as E_g and A_1g. Both Raman mode frequencies harden with pressure. At 18 GPa, the doubly degenerate mode E g splits, followed by softening of one of the split components. This is also accompanied by rapid softening of the high frequency mode. Features observed across 18 GPa con firm the reported structural phase transition. Beyond 23 GPa, there is an abrupt loss of Raman intensity and no Raman spectra could be recorded. This could be due to another structural transition. Raman spectroscopic data in the first high pressure phase give evidence for rearrangement in the FeO₆ octahedra which eventually leads to an increased coordination of copper in the second high pressure phase. Using the temperature dependence and the present pressure dependence of Raman mode frequencies of CuFeO₂, implicit and explicit contributions to their anharmonicity in the ambient phase are separated. The Raman spectroscopic data are correlated with available structural data for the delafossite  family of compounds.