Sixty years ago Paul Halmos concluded his Lectures on Ergodic Theory with a chapter Unsolved Problems which contained a list of ten problems. I will discuss some of these and some of the work that has been done on them. He considered actions of $\mathbb{Z}$ but I will also widen the scope to actions of general countable groups.

While most of Christian's work was in number theory he made important contributions to several aspects of ergodic theory throughout his career. I will discuss some of these and their impact on later developments.

I will discuss a criterion for randomness of sequences of zeros and ones which is strictly stronger than normality, butholds for almost every sequence generated by i.i.d. random variables with distribution {1/2, 1/2}. Briefly put, the idea is count the number of times blocks of length n appear in the initial block of length $2^n$. I will also discuss an extension of this idea to toral automorphisms. (joint work with Yuval Peres)