The th Ramanujan prime is the smallest positive integer such that if , then the interval contains at least primes. We sharpen Laishram's theorem that by proving that the maximum of is . We give statistics on the length of the longest run of Ramanujan primes among all primes , for . We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. An appendix explains Noe's fast algorithm for computing .