We review the idea of the margin of a classifier, and why that may be a good criterion for measuring a classifier's desirability. Then we consider the computational problem of finding the largest margin linear classifier. At this point we look at our toes with embarassment and note that we have only done work applicable to noise-free data. But we cheer up and show how to create a noise resistant classifier, and then a non-linear classifier. We then look under a microscope at the two things SVMs are renowned for---the computational ability to survive projecting data into a trillion dimensions and the statistical ability to survive what at first sight looks like a classic overfitting trap.