The modern S-matrix program has seen immense progress constructing scattering amplitudes in gauge and gravity theories without the aid of a Lagrangian, instead using only physical principles like Lorentz invariance and factorization. In this talk, I discuss how a similar approach is sufficient to construct effective field theories (EFTs), yielding a ``periodic table'' of scalar EFTs labelled by two numbers. We derive these EFTs--Lagrangians, symmetries, and all--from an ansatz of all possible S-matrices consistent with Lorentz invariance, factorization, and a certain infrared property, obtaining well-known examples like the chiral Lagrangian, Dirac-Born-Infeld theory, and the Galileon. We the show how tree amplitudes in these EFTs are constructible via on-shell recursion relations. With these methods one can prove stringent no-go theorems on the space of EFTs and, in principle, even discover new ones.