ABSTRACT The calculation of nestedness has become a routine analysis in the study of ecological networks, as it is commonly associated with community resilience, robustness and species persistence. While meaningful in species distributional patterns, for an interaction matrix to be nested, specialist species must interact with ordered subsets of subsequently more generalized species — not just with a lower number of species. However, after reviewing 419 papers on mutualistic networks published since nestedness was introduced for the study of species interactions in 2003, we have found that only two theoretical studies considered explicitly ordered subsets. Instead, most studies interpret nestedness as a core of densely connected generalist species, surrounded by a periphery of specialist species attached to this core — a so-called core-periphery structure. Such a topological feature is generally perceived as a core-periphery structure in network science. Here, we argue that the concept of core-periphery may be more relevant for studies on mutualistic networks than the concept of nestedness, as ecologists are usually not interested in exploring in detail the ordered subsets that characterize nestedness but instead use nestedness to describe a topology with a core of densely linked generalist species surrounded by a sparsely linked periphery of specialists. To illustrate our arguments and the quantification of core-periphery structures, we calculate core-periphery and nestedness in a large publicly available dataset of mutualistic networks. We also describe the calculation of core-periphery structures, its relationship with nestedness, and provide the code inside the R package econetwork for its calculation in mutualistic networks. We hope that our review will help ecologists to move beyond nestedness towards a more explicit representation of the structure of ecological networks.