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In 2015, Bryant, Horsley, Maenhaut, and Smith, generalizing a well‐known conjecture by Alspach, obtained the necessary and sufficient conditions for the decomposition of the complete multigraph λKn−I into cycles of arbitrary lengths, where I is empty, when λ(n−1) is even and I is a perfect matching, when λ(n−1) is odd. Moreover, Bryant in 2010, verifying a conjecture by Tarsi, proved that the obvious necessary conditions for packing pairwise edge‐disjoint paths of arbitrary lengths in λKn are also sufficient. In this article, first, we obtain the necessary and sufficient conditions for packing edge‐disjoint cycles of arbitrary lengths in λKn−I. Then, applying this result, we investigate the analogous problem of the decomposition of the complete uniform multihypergraph μKn(k) into Berge cycles and paths of arbitrary given lengths. In particular, we show that for every integer μ≥1, n≥108 and 3≤k