There are two significant challenges for analysts conducting economic evaluations of advanced therapy medicinal products (ATMPs): (i) estimating long-term treatment effects in the absence of mature clinical data, and (ii) capturing potentially complex hazard functions. This review identifies and critiques a variety of methods that can be used to overcome these challenges. The narrative review is informed by a rapid literature review of methods used for the extrapolation of survival analyses in the economic evaluation of ATMPs. There are several methods that are more suitable than traditional parametric survival modelling approaches for capturing complex hazard functions, including, cure-mixture models and restricted cubic spline models. In the absence of mature clinical data, analysts may augment clinical trial data with data from other sources to aid extrapolation, however, the relative merits of employing methods for including data from different sources is not well understood. Given the high and potentially irrecoverable costs of making incorrect decisions concerning the reimbursement or commissioning of ATMPs, it is important that economic evaluations are correctly specified, and that both parameter and structural uncertainty associated with survival extrapolations are considered. Value of information analyses allow for this uncertainty to be expressed explicitly, and in monetary terms.