It is widely believed that each fixed irreducible polynomial with no fixed prime factors can capture infinitely
many prime values. Moreover, the celebrated Schinzel Hypothesis asserts that any tuple of irreducible
polynomials without obvious obstructions can take prime values simultaneously infinitely often.
In this talk, we will give a survey on the case of linear and quadratic polynomials, and present relevant
tools developed throughout analytic number theory, including sieve method, circle method, estimate for
algebraic/analytic exponential sums.