Part 1 of this paper has discussed the uncertainties arising from gaps in knowledge or limited understanding of the processes involved in different natural hazard areas. Such deficits may include uncertainties about frequencies, process representations, parameters, present and future boundary conditions, consequences and impacts, and the meaning of observations in evaluating simulation models. These are the epistemic uncertainties that can be difficult to constrain, especially in terms of event or scenario probabilities, even as elicited probabilities rationalized on the basis of expert judgements. This paper reviews the issues raised by trying to quantify the effects of epistemic uncertainties. Such scientific uncertainties might have significant influence on decisions made, say, for risk management, so it is important to examine the sensitivity of such decisions to different feasible sets of assumptions, to communicate the meaning of associated uncertainty estimates, and to provide an audit trail for the analysis. A conceptual framework for good practice in dealing with epistemic uncertainties is outlined and the implications of applying the principles to natural hazard assessments are discussed. Six stages are recognized, with recommendations at each stage as follows: (1) framing the analysis, preferably with input from potential users; (2) evaluating the available data for epistemic uncertainties, especially when they might lead to inconsistencies; (3) eliciting information on sources of uncertainty from experts; (4) defining a workflow that will give reliable and accurate results; (5) assessing robustness to uncertainty, including the impact on any decisions that are dependent on the analysis; and (6) communicating the findings and meaning of the analysis to potential users, stakeholders, and decision makers. Visualizations are helpful in conveying the nature of the uncertainty outputs, while recognizing that the deeper epistemic uncertainties might not be readily amenable to visualizations.