Statistical Inference and Bayesian Methods
I first really encountered statistical ideas studying mathematics at Oxford. In my second year I took a course on statistical inference. It was basically a course on linear regression with a few more general bits on estimation and hypothesis testing. We did not encounter a single data set: it was all probability theory and huge amount of matrix algebra. I completed it, convinced that ANOVA was a Russian mathematician! Then in my third year I encountered Bayesian Statistics and suddenly things made sense to me. I have been committed to the Bayesian School ever since. My doctoral research was on Bayesian methods in protein crystallography, and I was among the vanguard applying Kalman filtering to solve hierarchical normal models. There was no MCMC methods to make life 'easy' and we spent hours squeezing the analysis into the available computing power. Keith Wilson and I recently wrote a retrospective on one application of Bayesian methods in crystallography (link) which has most certainly stood the test of time. It is still used and was instrumental in bringing Bayesian Statistics to the attention of the crystallographic community.
On completing my doctorate I moved away from applications of statistics toward decision and risk analysis. I have maintained a strong interest in statistical methodology though and have taught the subject over the years to many groups. My advocacy of the Bayesian School is well summarised in David Rios Insua's and my book on Statistical Decison Analysis 
 S. French and D. Rios Insua (2000) Statistical Decision Theory. Edward Arnold, London . Now published by Wiley. Details.