This paper proposes the co-prime circular microphone array (CPCMA), which applies co-prime array theory to circular microphone arrays. Compared with a conventional uniform circular array (UCA) with the same number of microphones and radius, the CPCMA achieves beampatterns with a narrower main lobe and fewer side lobes, whilst also avoiding spatial aliasing, thus having no grating lobes above the Nyquist frequency of the UCA. Array gain (AG) results indicate that the CPCMA is better than the UCA at amplifying the desired target signal while suppressing noise from other directions for typical speech signal frequencies. Compared with a UCA with similar performance and the same spatial Nyquist frequency, the CPCMA significantly reduces the required number of microphones. Simulations also indicate advantages of the CPCMA in speech source DOA estimation under high noise and low reverberation, especially when there are simultaneous multiple sources.