These lecture notes are based on lectures given by the authors at the autumn school
“Computations in motivic homotopy theory” at Regensburg University during
September 16–20, 2019. Main results include a computation of the first Milnor–Witt
stem of stable homotopy groups of motivic spheres over a field, presented differently
than Röndigs, Spitzweck and Østvær (2019), and a partial computation of the
zeroth Milnor–Witt stem of stable homotopy groups of motivic spheres over
discrete valuation rings of mixed characteristic after inverting the residue
characteristic.