The magnetism of Co1-xMnx alloys strongly depends on their structure and the Mn content.An ab initio linearized augmented plane wave (LAPW)method has been carried out to provide the electronic and magnetic structures for the fcc and bcc Co1-xMnx alloys with x=0.00,0.25,0.50,0.75 and 1.00. By increasing the Mn content,the magnetism of fcc Co1-xMnx alloys changes from ferromagnetism to ferrimagnetism,finally to antiferromagnetism;on the other hand, the magnetism of bcc Co1-xMnx alloys changes from ferromagnetism to ferrimagnetism.A good agreement is reached between out calculations and the previous experimental results.