In this dissertation, we study the field patching technique and its application to local-global principles for various objects. We first partially generalize the field patching technique initially proposed by Harbater-Hartmann to Hensel semi-global fields, i.e., function fields of curves over excellent henselian discretely valued fields. More specifically, we prove that patching holds for torsors under finite constant groups. Within this new framework, we further study local-global principles for principal homogeneous spaces as well as for higher degree Galois cohomology groups over Hensel semi-global fields. As an application, we study the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields. On a different perspective, we also extend some of these works to differential objects. In particular, we systematize Picard-Vessiot theory using differential torsors for algebraic groups and study local-global principles under differential objects.