Abstract: There have been several recent approaches to defining a moduli space of Langlands parameters (or L-parameters) over the integers, in order to obtain refined versions of the local Langlands conjecture "at all primes away from p at once". I will discuss the approach of Dat-Helm-Kurinczuk-Moss, including the basic results and the example of split GL_2. There is a conjectural description of the connected components of this moduli space, of which I will outline a proof if time permits.