The classical Titchmarsh divisor problem asks for the asymptotic evaluation of the divisor function over shifted primes. It is intimately related with primes in long arithmetic progressions. Modern methods can produce strong error terms for fixed shifts, but no progress since the 1960s has been made on the dual problem of summing d(n-p) for p < n or the related problem of Hooley and Linnik of representing a number a sum of a prime and two squares. I will survey approaches and techniques towards the Titchmarsh divisor problem and its variations, and present new results obtained in joint work with Edgar Assing and Junxian Li. The methods involve a blend of classical analytic number theory, automorphic forms and algebraic geometry.