We discuss the representation of primes, almost-primes, and related arithmetic sequences as sums of kth powers of natural numbers. In particular, we show that on GRH, there are infinitely many primes represented as the sum of 2\lceil 4k/3\rceil positive integral kth powers, and we prove unconditionally that infinitely many P_2-numbers are the sum of 2k+1 positive integral kth powers. The sieve methods required to establish the latter conclusion demand that we investigate the distribution of sums of kth powers in arithmetic progressions, and our conclusions here may be of independent interest.