The flux-corrected-transport paradigm is generalized to finite element schemes based on arbitrary time-stepping. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory low-order method is constructed by elimination of negative off-diagonal entries of the discrete transport operator. The linearization of source terms and extension to hyperbolic systems are discussed. Zalesak`s multidimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof of positivity is provided. The treatment of nonlinearities and iterative solution of linear systems are addressed. The performance of the new algorithm is illustrated by numerical examples for the shock tube problem in 1D and scalar transport equations in 2D.