The Sums-of-AM/GM-Exponential (SAGE) approach to signomial and polynomial nonnegativity is a powerful proof system based on convex duality and the relative entropy function. In this talk I provide a brief derivation of SAGE certificates for signomials, and review some recent results for SAGE signomials and SAGE polynomials. I will conclude with a small preview of upcoming work: how the "signomial representatives" underlying SAGE polynomials lead to an effective algorithm for solution recovery to moment-SAGE relaxations in polynomial optimization.