Abstract: A $\Pi$-algebra is the formal analog of the graded homotopy groups $ \pi_*X$ for a pointed topological space, together with its primary homotopy operations. I'll discuss joint work with David Blanc and Jim Turner, which began with constructing an obstruction theory to detect when diagrams of Pi-algebras arise by applying $\pi_*$ to a commutative diagram of pointed spaces. These obstructions lie in certain Andre-Quillen cohomology groups, and our recent work has focused on identifying ``the last obstruction" to realizing finite, directed diagrams with a general definition of higher homotopy operation. As one example, Toda brackets arise by applying $\pi_*$ to a homotopy commutative diagram and then looking for an obstruction to finding a strictly commutative model, where the null composites are replaced by actual basepoint maps. We have also developed new computational tools for the Andre-Quillen cohomology of diagrams, building three types of spectral sequences from towers of fibrations between spaces of natural transformations. In the hopes of making all of this machinery more palatable, I'll also try to say a few words about where we hope to go next.