The classical investment models like the famous Markowitz Model are one-period models based only on the expected growth rate and volatility of a stock, and do not make any assumptions on the exact behaviour or probability distributions of risky assets. It was a milestone of Robert C.Merton to consider an investment-consumption problem where risky assets follow a Geometric Brownian Motion. He derives that the investment decision is independent of the up and down movement of the stock, as it is optimal to always hold the same proportion of wealth invested in risky assets. As soon as the investor is faced with transaction costs however, he must match the benefits of improved diversification against the associated transaction costs. This book tries to outline several important theories and results concerning proportional transaction costs.