Despite being such an important figure in the New Testament, scholars are not sure where he preached the Gospel (both Scythia and Epirus in Greece claimed him as their apostle), where he died or even where he was buried. However, the manner of his death is very well-documented.
Like most saints, a number of legends that have grown up about his life and holy work. One of these, regarding a journey to Ethiopia, is told in the Old English poem Andreas. But none of this explains how he came to be the patron saint of Scotland.
You can learn more about Saint Andrew and the Scottish city of St Andrews (along with its world famous university) in Scottish Miscellany: Everything You Always Wanted to Know About Scotland the Brave.