The quantitative mapping of food web flows based on empirical data is a crucial yet difficult task in ecology. The difficulty arises from the under-sampling of food webs, because most data sets are incomplete and uncertain. In this article, we review methods to quantify food web flows based on empirical data using linear inverse models (LIM). The food web in a LIM is described as a linear function of its flows, which are estimated from empirical data by inverse modeling. The under-sampling of food webs implies that infinitely many different solutions exist that are consistent with a given data set. The existing approaches to food web LIM select a single solution from this infinite set by invoking additional assumptions: either a specific selection criterion that has no solid ecological basis is used or the data set is artificially upgraded by assigning fixed values to, for example, physiological parameters. Here, we advance a likelihood approach (LA) that follows a different solution philosophy. Rather than singling out one particular solution, the LA generates a large set of possible solutions from which the marginal probability density function (mPDF) of each flow and correlations between flows can be derived. The LA is exemplified with an example model of a soil food web and is made available in the open-source R-software. Moreover, we show how stoichiometric data, stable isotope signatures, and fatty acid compositions can be included in the LIM to alleviate the under-sampling problem. Overall, LIM prove to be a powerful tool in food web research, which can bridge the gap between empirical data and the analysis of food web structures.