Tensor analysis has reached a celebrity status in the areas of machine learning, computer vision, and artificial intelligence. Completing and recovering tensor is an important problem for tensor analysis. It involves recovering a tensor from either a subset of its entries or the whole entries contaminated by noise. Classical tensor completion/recovery methods are based mainly on l2-norm (Gaussian distribution noise) models, and are sensitive to noise of large magnitude. While l1-norm (Laplacian distribution noise) based methods are robust to noise of large magnitude, they do not deal with dense noise effectively. In this paper, we present a novel Cauchy tensor decomposition method for simultaneously recovering and completing a low rank tensor with both missing data and complex noise. We utilize the Cauchy distribution to model noise and derive the objective function of Cauchy tensor decomposition under the maximum likelihood estimation (MLE) framework. Then we developed a robust tensor decomposition framework that used first-order optimization approaches to optimize the objective function. Extensive numerical experiments are conducted and show that our method is able to successfully recover and complete tensors with large/dense noise and missing data. We further demonstrate the usefulness of Cauchy tensor decomposition on three real-world applications, image inpainting, traffic data process and foreground/background separation. The experimental results show that the method is applicable to a wide range of problems.