The Manin-Mumford and Mordell-Lang conjectures (now theorems of Raynaud and Faltings, respectively) are among the cornerstones of modern arithmetic geometry. We will introduce natural dynamical analogs of each and present a variety of theorems as well some counterexamples. Roughly speaking, there is a natural one-parameter analog of Mordell-Lang that is true under some reasonable hypotheses, while the obvious dynamical analog of Manin-Mumford fails under the same hypotheses. We will explain why this is perhaps not surprising.