Abstract: The talk treats joint work of Yongwei Yao and the speaker that is in progress. Let $R$ be a Noetherian ring, let $P \in \Spec(R)$, and let $A:= R/P$. We discuss a theory of generically Artinian modules for $R$ at $P$ when $P$ is a prime that need not be maximal: results similar to Matlis duality hold on a Zariski neighborhood of $P$. We introduce the notion of a generically Artinian module at $P$, of a generically injective module at $P$, and of a generically injective hull $E$ for $R/P$. When it exists, $E$ turns out to be unique up to non-unique isomorphism after possibly passing to a smaller Zariski neighborhood of $P$. The results are proved under mild conditions on $R$. The key results show that many statements from classical duality theory hold after localizing at just one element of $R \setminus P$. Here is one example. If $H$ is any generically Artinian module at $P$ then, after localizing at one element $g \in R \setminus P$, the associated graded module of $H_g$, namely, $\bigoplus_{t = 0} ^\infty {{\Ann_{H_g}P^{t+1}}\over{\Ann_{H_g} P^t}}$, is free over $A_g$: in fact, all of its graded components are $A_g$-free. This parallels classical results of Grothendieck on generic freeness in EGA, but the strength of this and several other results is surprising, because the modules considered typically have neither ACC nor DCC. It turns out that under mild conditions on $R$, the local cohomology modules $H^i_P(M)$ for a Noetherian $R$-module $M$ are generically Artinian. These results recover, in a much more general framework, earlier work of the authors, which generalized results of Karen Smith and J\'anos Koll\'ar. The authors have used these ideas to settle long standing questions in tight closure theory.