Abstract : Two important players in enumerative geometry and mirror symmetry are the Fukaya categories of symplectic manifolds and the Donaldson-Thomas invariants of algebraic Calabi-Yau threefolds. I will give an introduction to these stories, and explain in what sense the latter can be regarded as a "shift" of the former. More generally, I will speculate on the existence of natural invariants of "n-shifted" symplectic spaces that live in category level n+1.