I will start out by saying I am not well-informed about moduli theory in the slightest. However, it is known that some moduli spaces (of algebro-geometric objects) have severe pathologies (Ex: Murphy's Law with regards to Hilbert schemes), while others are very nice, such as $\overline{\mathcal{M}}_{1,1}$ which can be identified with the famous modular surface, and on such things differential geometry can be done. This leads me to ask (whether anyone has already asked):

1) When does a moduli space (parametrizing algebro-geometric objects) admit natural Riemannian structure(s) besides possibly on some singular locus?

2) In this case, do geodesics correspond to useful information (nice degenerations between objects perhaps)?