There are three aims of this note. The first one is to report some advances around the dynamical Mordell-Lang ( \(=\) DML) conjecture. Second, we generalize some known results. For example, the Dynamical Mordell-Lang conjecture was known for endomorphisms of \(\mathbb {A}^2\) over \(\overline{\mathbb {Q}}\) . We generalize this result to all endomorphisms of \(\mathbb {A}^2\) over \(\mathbb {C}\) . We generalize the weak DML theorem to a uniform version and to a version for partial orbit. Using this, we give a new proof of the Kawaguchi-Silverman-Matsuzawa upper bound for arithmetic degree. We indeed prove a uniform version which works in both number field and function field case in any characteristic and it works for partial orbits. We also reformulate the “p-adic method”, in particular the p-adic interpolation lemma in language of Berkovich space and get more general statements. The third aim is to propose some further questions.