In this paper we report on recent results and applications of the so-called SG calculus, focusing, in particular, on its elliptic elements. The topics we discuss include propagation of singularities of (global versions of) wave-front sets, global regularity of solutions to linear and nonlinear partial differential equations on $\R^{d}$ in appropriate function and distribution spaces, and spectral asymtptotics of elliptic, selfadjoint, positive operators on $\R^{d}$.