Over the last two decades, tensor-based methods have received growing attention in the signal processing community. In this work, we propose a comprehensive overview of tensor-based models and methods for multisensor signal processing. We present for instance the Tucker decomposition, the Canonical Polyadic Decomposition (CPD), the Tensor-Train Decomposition (TTD), the Structured TTD, including Nested Tucker Train (NTT), as well as the associated optimization strategies. More precisely, we give synthetic descriptions of state-of-art estimators as the Alternating Least Square (ALS) algorithm, the High-Order SVD (HOSVD), and of more advanced algorithms as the Rectified ALS, the TT-SVD/TT-HSVD and the Joint dImensionally Reduction And Factor retrieval Estimator (JIRAFE) scheme. We illustrate the efficiency of the introduced methodological and algorithmic concepts in the context of three important and timely signal processing-based applications: the Direction-Of-Arrival (DOA) estimation based on sensor arrays, multidimensional harmonic retrieval and MIMO wireless communication systems.