In this talk I will discuss work with Frauke Bleher and Bart de Smit 
on the inverse problem for universal deformation rings.  This problem 
is to determine which complete Noetherian rings R with 
residue field of characteristic p occur as the universal
deformation rings of representations of profinite groups.
M. Flach asked whether for all p a ring R occurs which
is not a local complete intersection.  We show this 
is the case by proving that Z_p[[t]]/(p^n t,t^2) occurs
for all n > 0.  We actually know of no R as above
which does not occur.  I will also discuss the
so-called inverse inverse problem, in which one
asks for a given R for all groups and representations
giving rise to R as a universal deformation ring.