Constant-temperature molecular dynamics simulations and the Gupta-type many-body interatomic potential are used to study the melting behaviors of Co N ( N = 13, 14, 38, 55, 56) clusters. The atomic equivalence indexes are used to describe the structural changes of clusters at different temperatures. Other quantities like the Lindemann index, the heat capacity and caloric curve are also employed. The lag phenomena (the rapid change of Lindemann index is not consistent with that of the heat capacity curve during the cluster melting) are discussed in detail for the Co 13 and Co 14 clusters. Our results show that the Lindemann index fails to capture the melting transition while the heat capacity does for the Co 13 or Co 14 clusters. All these clusters show premelting behaviors and the surface melting are found for the Co 14 and Co 56 clusters. The Co 38 cluster undergoes distinct isomerization transitions prior to its full melting.