The classical transportation problem (having source and destination as indices) deals with the objective of minimizing a single criterion, i.e. cost of transporting a commodity. Additional indices such as commodities and modes of transport led to the Multi Index transportation problem. An additional fixed cost, independent of the units transported, led to the Multi Index Fixed Charge transportation problem. Criteria other than cost (such as time, profit etc.) led to the Multi Index Bi-criteria transportation problem. The application of fuzzy and stochastic concept in the above transportation problems would enable researchers not only to introduce real life uncertainties but also to obtain solutions of these transportation problems. The review article presents an organized study of the Multi Index transportation problem and its fuzzy and stochastic extensions till today, and aims to help researchers working with complex transportation problems.