The theory of harmonic Maass forms and mock modular forms has seen an explosion of activity in the past 20 years, with applications to physics, partitions, enumerative geometry, and many other topics. Along the way, much has been developed in the theory of harmonic Maass forms. However, until recently, harmonic Maass form theory lacked analogues of key structures that exist for classical holomorphic modular forms and Maass waveforms, such as the theory of L-functions. Recent work with Diamantis, Lee and Raji will be described which gives the first general such theory. In particular, I will sketch Weil-type Converse Theorems and a Voronoi-type summation formula in these settings. I will also describe connections with the construction of differential operators on these spaces and a more thorough explanation of a previous formula for a central L-value of the j-invariant, which had been discovered heuristically by Zagier and proven in that case by Bruinier, Funke, and Imamoglu.