We clarify the status of log-periodicity associated with speculative bubbles preceding financial crashes. In particular, we address Feigenbaum's criticism ( [A article="1469-7688/1/3/306"] Feigenbaum J A 2001 Quantitative Finance1 346-60 [/A] ) and show how it can be refuted. Feigenbaum's main result is as follows: 'the hypothesis that the log-periodic component is present in the data cannot be rejected at the 95% confidence level when using all the data prior to the 1987 crash; however, it can be rejected by removing the last year of data' (e.g. by removing 15% of the data closest to the critical point). We stress that it is naive to analyse a critical point phenomenon, i.e., a power-law divergence, reliably by removing the most important part of the data closest to the critical point. We also present the history of log-periodicity in the present context explaining its essential features and why it may be important. We offer an extension of the rational expectation bubble model for general and arbitrary risk-aversion within the general stochastic discount factor theory. We suggest guidelines for the use of log-periodicity and explain how to develop and interpret statistical tests of log-periodicity. We discuss the issue of prediction based on our results and the evidence of outliers in the distribution of drawdowns. New statistical tests demonstrate that the 1% to 10% quantile of the largest events of the population of drawdowns of the NASDAQ composite index and of the Dow Jones Industrial Average index belong to a distribution significantly different from the rest of the population. This suggests that very large drawdowns may result from an amplification mechanism that may make them more predictable.