Almost-primes represented by quadratic polynomials

Dirichlet's theorem on primes in arithmetic progressions characterizes those linear polynomials which take on prime values infinitely often. However, this is where the current state of knowledge ends. For the case of polynomials with higher degrees, heuristic arguments lead us to believe that for an irreducible polynomial with integer coefficients, if the leading coefficient is positive and the polynomial has no fixed prime divisor, then the polynomial represents primes infinitely often. I will discuss the case for quadratic polynomials with an emphasis on the work of Iwaniec.