We conducted a thorough experimental and numerical study of the micromagnetic properties of Permalloy (Ni80Fe20) microdisks exhibiting target domain structures at remanence. Vortex configurations are quite common in such microdisks and correspond to an in-plane flux closure configuration of cylindrical symmetry with an out-of-plane magnetized core. In contrast, target domain configuration are observed in thicker microdisks and are characterized by a vortex configuration of the in-plane component of the magnetization superposed to an out-of-plane component of magnetization which oscillates as a function of the distance to the microdisk center resulting in the formation of concentric domains. The ratio of the out-of-plane oscillatory component of the magnetization to the in-plane vortex one increases with the thickness of the microdisk. Hysteresis loops were measured under in-plane and out-of-plane field. The results at remanence and under magnetic field could be interpreted by micromagnetic simulations in which the microdisks were described as an assembly of partially coupled columns representing the granular nanostructure of the films from which the microdisks were patterned. Quite original magnetization processes take place in these microdisks exhibiting target domain remanent configuration. These include in particular entire flipping of the domain configuration and annihilation/creation of ring domains.