Automatic sequences and their number theoretic properties have been in-tensively studied during the last 20 or 30 years. Since automatic sequences are quiteregular (they just have linear subword complexity), they cannot be used as quasi-random sequences. However, the situation changes drastically when one uses propersubsequences, for examplethe subsequence along primesor squares. It isconjecturedthat the resulting sequences are normal sequences which could be already proved forthe Thue–Morse sequence along the subsequence of squares.This kind of research is very challenging and was mainly motivated by the Gelfondproblems for the sum-of-digits function. In particular, during the last few years spec-tacular progress was made due to the Fourier analytic method by Mauduit and Rivat.In this chapter, we survey these recent developments, comment on the proof methods,and formulate quite general conjectures. We also present a new result on the subse-quence along primes of so-called invertible automatic sequences.