Progressions. Dirichlet famously proved that, if $a$ and $b$ are coprime integers, the arithmetic progression $a+db$ (i.e., with $d=1,2,3,\dots$) contains infinitely many prime numbers. Are there any other such results? I believe there are no analogous quadratic or higher-power sequences, but I could have missed some.
Sequences. Other than the obvious/trivial ones (or arithmetic progressions, covered above), which number sequences [with a regular algebraic definition] contain infinitely many prime numbers?