My thesis focuses on computations of stable homotopy groups of spheres, with applications and connections to differential geometry and motivic homotopy theory. The Adams spec- tral sequences and Toda brackets play a major role in my work. We have introduced two methods to compute Adams differentials and solve extension problems: one is very technical but inductive, using the algebraic Kahn-Priddy theorem; the other one is more systematic, using a new connection between motivic homotopy theory and chromatic homotopy theory. Combining both methods, we have computed stable stems into a larger range. As a consequence, I solved the strong Kervaire invariant problem in dimension 62 and showed that the 61-sphere has a unique smooth structure, which is the last odd dimensional case.