Abstract: Distributional tensor analysis on manifolds has certain drawbacks, in particular a distributional curvature tensor cannot be defined in general, as this would involve ill-defined products of distributions. An established way to overcome such difficulties lies in the theory of Colombeau algebras, which enable one to perform nonlinear operations with distributions while retaining much consistency with the classical theory. In my talk I will first give a short summary of distributional tensor fields and generalized functions on a manifold and based on this I will then motivate and sketch the construction of an algebra of generalized tensor fields. I will skip technicalities and focus on desired key properties like (canonical) embeddings of smooth and distributional tensor fields, a definition of Lie derivative and covariant derivative for generalized tensor fields, and a generalized curvature tensor. I will conclude with a mention of the obstacles in obtaining these properties and possible ways around them.