In recent years physicists have become involved in studying the financial market and the vast data it generates. The constantly-updated streams of information are a perfect testing ground for the hypothesis that the laws of statistical physics might apply to human behaviour. In this thesis, I study two empirical log return time series for the stylised facts of financial data. I then use Multifractal Detrended Fluctuation Analysis to study the empirical log returns for multifractal scaling. I find that extreme events are inimical to the scaling in highly leptokurtic data. I also find that the temporal correlations in the data are crucial to the scaling whereas the shape of its distribution is not as important. I then develop my own agent-based model of the market. With just a few different types of traders operating according to some simple rules, my model generates log returns with many of the statistical properties found in empirical data. The option for traders to opt out of trading is the source of the thin-peaked distribution of the simulated log returns. The distribution of log returns becomes more closely described by a Gaussian at longer lags. This is a consequence of basing the fundamental value of the stock on geometric Brownian motion. Since transition to Gaussianity at long lags is also a feature of empirical log returns, this implies that real traders are also influenced by some geometric Brownian process. Log returns generated by the model also have volatility clustering, are uncorrelated and asymmetrically distributed. I test the log returns generated by my model for their scaling properties and find that they do not have multifractal scaling. This is an interesting result since the simulated log returns do feature other properties of empirical data. I then extend the model in some basic ways to include more heterogeneity. Some limited multifractal scaling is found in the simulated log returns of the extended model. Because the model produces stochastic output, it is extremely difficult to exactly determine the scaling properties. However the results hint at the possibility that the multifractality found in empirical log returns is a consequence of the heterogeneity in both the investment horizons and beliefs of traders in the market.