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By using the concept of tensor shape partition, our results include, as a special case, a wide variety of particular tensor spectral problems considered in the literature and can be applied to a broad set of problems involving tensors (and matrices), including the computation of operator norms, graph and hypergraph matching in computer vision, hypergraph spectral theory, higher-order network analysis, and multimarginal optimal transport. The key to our approach is to recast the eigenvalue problem as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multihomogeneous order-preserving maps to derive new and unifying Perron--Frobenius theorems for nonnegative tensors, which either imply earlier results of this kind or improve them, as weaker assumptions are required. We introduce a general power method for the computation of the dominant tensor eigenpair and provide a detailed convergence analysis. This paper is directly based on our previous work [A. Gautier, F. Tudisco, and M. Hein, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1206--1231] and complements it by providing an extended introduction and several new results.<\/jats:p>","DOI":"10.1137\/23m1557489","type":"journal-article","created":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T10:05:24Z","timestamp":1683540324000},"page":"495-536","source":"Crossref","is-referenced-by-count":10,"title":["Nonlinear Perron--Frobenius Theorems for Nonnegative Tensors"],"prefix":"10.1137","volume":"65","author":[{"given":"Antoine","family":"Gautier","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8150-4475","authenticated-orcid":true,"given":"Francesco","family":"Tudisco","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthias","family":"Hein","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,5,9]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/19M1269713"},{"key":"atypb2","first-page":"20190724","volume":"476","author":"Arrigo F.","year":"2020","journal-title":"Proc. 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