We shall review some of  the general problems which are unsolved on rigid local systems and arithmetic  -adic local systems. We ‘ll report briefly on a proof (2018 with Michael Groechenig) of Simpson's integrality conjecture for  . While all rigid local systems in dimension are cohomologically rigid (1996, Nick Katz), we did not know until last week  of a single example in higher dimension which is rigid but not cohomologically rigid. We’ll present one series of examples  (2022, joint with Johan de Jong and Michael Groechenig).