Detectors and electronics inside the ATLAS detector at the large hadron collider will be exposed to high fluxes of photons, charged particles, and neutrons. Damage caused by radiation will influence the performance of the detector. It will therefore be important to continuously monitor the radiation dose to follow the level of detector degradation. It will also be important to have information about the radiation field in the detector early in the life of the experiment in order to check the simulations and to correctly predict future radiation damage. An online radiation monitoring system with semiconductor radiation sensors will monitor the dose at several locations in the ATLAS detector. It will measure ionizing dose in SiO with RadFETs, displacement damage in silicon in terms of 1-MeV(Si) equivalent neutron fluence with p-i-n diodes and fluence of thermal neutrons from current gain degradation in dedicated bipolar transistors. The design of the system and its response to various types of radiation is described.