The homotopy groups of the motivic sphere spectrum are the invariants governing the universe of (cellular) stable motivic homotopy theory. I will motivate and construct this bi-graded system of groups over a general base field, and then delve into their structural aspects, focusing on vanishing lines and behavior in the eta-periodic range, where the motivic Hopf map acts as an isomorphism. I will conclude by discussing a slice spectral sequence approach to the homotopy groups of the eta-periodic motivic sphere (joint with Oliver Röndigs) which provides complete information in case the base field has odd characteristic or cohomological dimension at most one. This recovers a theorem of Andrews-Miller over C and suggests a "Witt-theoretic image of J pattern" in general.