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The discovery of correlated states and superconductivity in magic-angle twisted bilayer graphene (MATBG) has established moiré quantum matter as a new platform to explore interaction-driven and topological quantum phenomena. Multitudes of phases have been realized in moiré systems, but surprisingly, robust superconductivity has been one of the least common of all, initially found in MATBG and only more recently also in magic-angle twisted trilayer graphene (MATTG). While MATBG and MATTG share some similar characteristics, they also exhibit substantial differences, such as in their response to external electric and magnetic fields. This raises the question of whether they are simply two separate unique systems, or whether they form part of a broader family of superconducting materials. Here, we report the experimental realization of magic-angle twisted 4-layer and 5-layer graphene (MAT4G and MAT5G, respectively), which turn out to be superconductors, hence establishing alternating-twist magic-angle multilayer graphene as a robust family of moiré superconductors. The members of this family have flat bands in their electronic structure as a common feature, suggesting their central role in the observed robust superconductivity. On the other hand, there are also important variations across the family, such as different symmetries for members with even and odd number of layers. However, our measurements in parallel magnetic fields, in particular the investigation of Pauli limit violation and spontaneous rotational symmetry breaking, reveal that the most pronounced distinction is between the N=2 and N>2-layer structures. Our results expand the emergent family of moiré superconductors, providing new insight with potential implications for the design of novel superconducting materials platforms.