Dating from the seminal work of Nobel Laureates Markowitz and Merton, some very sophisticated optimization and stochastic control models have been developed for the purpose of effectively managing a portfolio of securities. Some of these are very sophisticated and appear to be very realistic. But aside from some possible use by some private hedge funds, these models are apparently rarely used in practice. This lecture will examine why. In the course of surveying the well known as well as some promising new mathematical approaches to portfolio management, and the various methods will be compared using modest backtests with market data. While there is reason to be encouraged about the use of mathematical optimization models for portfolio management, our main conclusion is that future research on portfolio optimization needs to be more cognizant of practicalities and the realities of financial markets.