Assimilation of observation data in meteorology and space weather consists of using these data to estimate the current state and the spatially and temporally distributed parameters of Numerical Weather Prediction (NWP) models, which are often fluid dynamical equations. The aim of the data assimilation is to provide wider monitoring of the weather condition beyond the locations where data are collected, also referred to as now-casting and to provide forecasting of weather conditions using NWP. The development of Global Assimilative Ionosphere Model (GAIM) started about 15 years ago as massive amount of ionosphere observations became available with the installation of global network of ground based GPS receivers, as well as, the launches of multiple space based GPS receivers for collecting Radio Occultation (RO) data. The development of GAIM has shown, for the first time, that space weather monitoring and forecast can be successful made using physics law based NWP. The outputs of GAIM are currently used in operational space weather reports and space weather research programs. However, several important mathematical challenges are still present in the assimilation of ionospheric observations. In this presentation, I shall provide an overview of the development of GAIM and the mathematical challenges the developers of GAIM are facing.