We find formulas for the determinants of several Toeplitz-Hessenberg matrices whose nonzero entries are Fuss-Catalan numbers. By Trudi's formula, one obtains equivalent multi-sum identities indexed over the set of partitions of a fixed integer involving the product of Fuss-Catalan numbers and multinomial coefficients. We make use of generating functions to establish our results and, as a consequence, several entries from the OEIS are afforded new combinatorial interpretations as determinants of certain Toeplitz-Hessenberg matrices. Finally, we provide counting arguments for several of our determinant identities drawing upon combinatorial interpretations of various specific cases of the Fuss-Catalan numbers in terms of Dyck paths, ternary trees, and non-crossing partitions.