Euler products inside the critical strip

Even though Euler products of L-functions are generally valid only to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip. Indeed, the behaviour of Euler products inside the critical strip is very closely related to several major problems in number theory including the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. In this talk, we will give an introduction to this topic and then discuss recent work on establishing asymptotics for partial Euler products of L-functions in the critical strip. We will end by giving applications of these results to questions related to Chebyshev's bias.