The thesis considers the general area of robust portfolio construction. In particular the thesis considers two techniques in this area that aim to improve portfolio construction, and consequently portfolio performance. The first technique focusses on estimation error in the sample covariance (one of portfolio optimisation inputs). In particular shrinkage techniques applied to the sample covariance matrix are considered and the merits thereof are assessed. The second technique considered in the thesis focusses on the portfolio construction/optimisation process itself. Here the thesis adopted the 'resampled efficiency' proposal of Michaud (1989) which utilises Monte Carlo simulation from the sampled distribution to generate a range of resampled efficient frontiers. Thereafter the thesis assesses the merits of combining these two techniques in the portfolio construction process. Portfolios are constructed using a quadratic programming algorithm requiring two inputs: (i) expected returns; and (ii) cross-sectional behaviour and individual risk (the covariance matrix). The output is a set of 'optimal' investment weights, one per each share who's returns were fed into the algorithm. This thesis looks at identifying and removing avoidable risk through a statistical robustification of the algorithms and attempting to improve upon the 'optimal' weights provided by the algorithms. The assessment of performance is done by comparing the out-of-period results with standard optimisation results, which highly sensitive and prone to sampling-error and extreme weightings. The methodology looks at applying various shrinkage techniques onto the historical covariance matrix; and then taking a resampling portfolio optimisation approach using the shrunken matrix. We use Monte-Carlo simulation techniques to replicate sets of statistically equivalent portfolios, find optimal weightings for each; and then through aggregation of these reduce the sensitivity to the historical time-series anomalies. We also consider the trade-off between sampling-error and specification-error of models.