Radiomics features extract quantitative information from medical images, towards the derivation of biomarkers for clinical tasks, such as diagnosis, prognosis, or treatment response assessment. Different image discretization parameters (e.g. bin number or size), convolutional filters, segmentation perturbation, or multi-modality fusion levels can be used to generate radiomics features and ultimately signatures. Commonly, only one set of parameters is used; resulting in only one value or flavour for a given RF. We propose tensor radiomics (TR) where tensors of features calculated with multiple combinations of parameters (i.e. flavours) are utilized to optimize the construction of radiomics signatures. We present examples of TR as applied to PET/CT, MRI, and CT imaging invoking machine learning or deep learning solutions, and reproducibility analyses: (1) TR via varying bin sizes on CT images of lung cancer and PET-CT images of head & neck cancer (HNC) for overall survival prediction. A hybrid deep neural network, referred to as TR-Net, along with two ML-based flavour fusion methods showed improved accuracy compared to regular rediomics features. (2) TR built from different segmentation perturbations and different bin sizes for classification of late-stage lung cancer response to first-line immunotherapy using CT images. TR improved predicted patient responses. (3) TR via multi-flavour generated radiomics features in MR imaging showed improved reproducibility when compared to many single-flavour features. (4) TR via multiple PET/CT fusions in HNC. Flavours were built from different fusions using methods, such as Laplacian pyramids and wavelet transforms. TR improved overall survival prediction. Our results suggest that the proposed TR paradigm has the potential to improve performance capabilities in different medical imaging tasks.