The practice of advanced statistics normally requires the use of sophisticated software, not due to theoretical complexity but rather to sheer volume and repetition of required calculations. Without computers, advanced statistical number crunching would be too time consuming, especially with large data sets.
A course in advanced statistics will most likely follow the experience and particular interests of the professor. It is safe to say, however, that whatever material is covered will most likely involve a sophisticated treatment of many of the following topics:
First and foremost, students should take advantage of MIT's Open Courseware that offers the highest quality advanced statistics courses online (for free), including Mathematical Statistics and Topics in Statistics: Nonparametrics and Robustness. Quite a selection of good books is available on advanced statistics, many of which can be found on Google Books and Amazon.com. Students should also follow Springer's Advances in Statistical Analysis and Advances in Data Analysis and Classification, as well as Pushpa Publishing House's Advances and Applications in Statistics, and Scientific Advances Publishers' Journal of Statistics: Advances in Theory and Applications.