Abstract: The Riemann Hypothesis, often considered one of the most important open problems in mathematics, would have a number of fantastic consequences for our understanding of the distribution of prime numbers. It claims that all the (non-trivial) zeros of a complex function (the Riemann zeta function) have real part equal to 1/2. However, it turns out that several of these consequences would actually follow from a weaker 'zero density' result that says 'most' zeros lie 'close' to the line with real part equal to 1/2. I'll describe this picture assuming no prior knowledge, as well as describing some recent work (joint with Larry Guth) that improves an 80-year-old estimate on the density of zeros, with corresponding improvements for the distribution of primes.