Data Assimilation methods integrate numerical models and measurement data to provide state (and parameter) estimates, including uncertainty estimates, that serve as initial values for real‐world forecasts or as a basis for monitoring and statistical analysis. Today, data assimilation is a well‐established field, with a broad range of methods ranging from variational or minimization approaches to ensemble data assimilation, such as Ensemble Kalman filters or particle filters. At the same time, the field sees intense and rapid developments on all its parts, ranging from new algorithms to improved treatment of spatially and/or temporally dense observations, satellite borne and ground‐based. The present special section arose from discussions and presentations at the International Symposium on Data Assimilation (ISDA2016). These international symposia of between 100 and 200 participants have been run with a frequency of 1–2 years since 2011 in Europe and Asia. Several survey papers have been composed by groups of leading scientists, complemented by more specific topics of current interest. Data assimilation methods today are addressing challenges of geophysical numerical models with higher and higher resolution and complexity, as well as vast amounts of complex, sometimes highly nonlinear observations. The higher resolution models with grid spacing on the order of kilometers are able to resolve strongly nonlinear dynamics and start to resolve physical processes that have traditionally been parameterized such as, for example, convection. The initialization of such models requires utilization of temporally and spatially dense observations. While modern ground‐ and satellite‐based remote sensing instruments provide a vast amount of high‐resolution observations, methods that appropriately use these data still need to be explored. Especially challenging are observations related to, or affected by clouds and precipitation. These observations are directly linked to the key forecast phenomena for high‐resolution numerical weather prediction models such as convective precipitation and severe wind gusts. Challenges for using such observations include the development of appropriate observation operators, assignment of observation errors including the representation error and the treatment of associated nonlinearities and non‐Gaussian error distributions. Further, these observations require a high updating frequency, and treatment of deficiencies in uncertainty quantification of the model, in particular for representation of clouds and hydrometeors that lead to systematic deviations between the model and observations. A major problem is that observations, such as satellite radiances or radar reflectivity, can be difficult to assimilate due to location and timing differences between the background (forecast) of the clouds or storms and their observed counterparts. This problem arises in both global operational as well as in convective‐scale data assimilation applications. We start with a review of current developments towards assimilating cloud and precipitation affected satellite radiances at operational forecasting centers (Geer et al., 2018). Here the so called all‐sky data‐assimilation approach, which assimilates all observations directly as radiances, is particularly in focus. The hope is that assimilating frequently available all‐sky infrared observations from geostationary satellites could give particular benefit for short‐range forecasting and improve the analysis and shorter‐range forecast of poorly observed weather phenomena as diverse as tropical cyclones and wintertime low cloud. In this special section, we also survey current state of the art techniques used for convective‐scale data assimilation at operational forecasting centers (Gustafsson et al., 2018). Convection permitting models partly resolve highly nonlinear dynamics and physics at a range of spatial and temporal scales. They are currently pushing the boundary of more frequent assimilation of data and for them a particular challenge is development of multi‐scale data‐assimilation methods. Model errors from global models as well as from unresolved scales and processes are difficult to take into account and quantify for convection permitting models. Still, the quality of forecasts based on initial data from convective‐scale data assimilation is significantly better than the quality of forecasts from simple downscaling of larger‐scale initial data. Advances in methods applied at convective scales provide improvements compared to simpler methods, motivating continued research and development in convective‐scale data assimilation. A third review is provided for the long‐standing problem of representation error that arises from a different representation of reality between models and observations (Janjić et al., 2018). Representation error dominates the errors that lead to large all‐sky departures from forecast. Its proper treatment is also necessary in order to extract more information from existing observing systems of convective‐scale phenomena, for example weather radar data. The review, in addition to a theoretical framework and examples from different geophysical application areas, discusses diagnosing representation‐error statistics as well as their use in state‐of‐the‐art data assimilation systems. It also includes a discussion of algorithmic modifications necessary for the data‐assimilation algorithms to take the representation error into account, as well as of the use of correlated observation errors in practice. It has become clear that proper observation‐error statistics are important in order to extract the information from the vast amount of available data and therefore estimating statistics of the representation error is beginning to receive much more research attention. What underscores some of the challenges facing data assimilation are state and observation errors that are non‐Gaussian in nature, causing a problem with classical data‐assimilation algorithms that use Gaussian error assumptions to produce initial condition estimates from the previous forecast and incoming data. Three papers in this special section introduce or improve on the current methodology. For example, Robert et al. (2018) show how particle filters can be combined with localized ensemble Kalman filter for convective‐scale data‐assimilation applications. Sakov et al. (2018) introduce a model error term into the algorithm that was designed for tackling nonlinearities in the dynamics of chaotic models. Finally, Zhu et al. (2018) discuss a method to estimate errors in the model equations, using state‐of‐the‐art particle filters. Data assimilation for geophysical models that resolve many scales of motion and for observations of higher temporal and spatial density requires re‐evaluating and improving the methodology that is currently inherited from less nonlinear applications. Improved data‐assimilation systems will therefore require the development of novel approaches that account for non‐Gaussianity, model errors, better understanding how different existing methods can represent uncertainty at the data assimilation and forecasting steps of both observations and numerical models. This special section gives an overview of the current standing of research and emphasizes open problems. We would like to thank our colleagues for the inspiring discussions and contributions in the symposia and for contributions to this special section. We would also like to thank the referees who provided their time to ensure the high standard of papers in this special section. We are looking forward to the next steps in the development of this exciting and very dynamic field of research with very important and high‐impact applications in a large number of weather services and numerous research institutions and business branches around the world.