A compact planar antenna is proposed to operate in five millimeter-wave frequency bands at 28, 51, 55, 59, and 63 GHz. The antenna is designed to produce circular polarization at 28 GHz and linear polarization at the other four frequencies. The 28 GHz frequency band is recommended for long range communications, whereas the other four frequencies are recommended for short range communications due to the atmospheric absorption in this range of the millimeter-wave frequencies and to provide more secure communications. The reasons behind the selection of the specific values of the higher frequency bands (51, 55, 59, and 63 GHz) are (i) to provide enough number of multiple bands at such unlicensed frequency range to be used as alternatives for different applications, (ii) to give about 4 GHz separation between the center frequencies of each band and, (iii) finally, to avoid the operation at 60 GHz, in particular, due to the peak atmospheric absorption of millimeter-wave encountered at this frequency. The antenna is constructed as a square patch surrounded by five parasitic elements that are capacitively coupled to the square patch so as to realize the location of the resonant frequencies of the higher frequency bands. The antenna is fabricated and its performance is experimentally evaluated. The proposed antenna is shown to operate efficiently over the five frequency bands 28, 51, 55, 59, and 63 GHz providing impedance matching bandwidths of 2.0, 0.9, 1.2, 1.1, and 1.6 GHz, respectively. The corresponding values of the maximum gain are 7.0, 6.8, 6.0, 6.0, and 8.0 dBi, respectively. Also, the corresponding radiation efficiencies are 90%, 90%, 91%, 78%, and 86%, respectively. The 3 dB- axial ratio bandwidth at 28 GHz is about 1.2 GHz.