The motivic Adams spectral sequence

We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed. field of characteristic 0. Our results are based on computer calculations and a motivic version of the May spectral sequence. We discuss features of the associated Adams spectral sequence and use these tools to give new proofs of some results in classical algebraic topology. We also consider a motivic Adams-Novikov spectral sequence. The investigations reveal the existence of some stable motivic homotopy classes that have no classical analogue.