In this study melting of irregular ice crystals was observed in an acoustic levitator. The evolution of the particle shape is captured using a high-speed video system. Several typical phenomena have been discovered: change of the particle shape, appearance of a capillary flow of the melted liquid on the particle surface leading to liquid collection at the particle midsection (where the interface curvature is smallest), and appearance of sharp cusps at the particle tips. No such phenomena can be observed during melting of spherical particles. An approximate theoretical model is developed which accounts for the main physical phenomena associated with melting of an irregular particle. The agreement between the theoretical predictions for the melting time, for the evolution of the particle shape, and the corresponding experimental data is rather good.