Particle therapy of moving targets is still a great challenge. The motion of organs situated in the thorax and abdomen strongly affects the precision of proton and carbon ion radiotherapy. The motion is responsible for not only the dislocation of the tumour but also the alterations in the internal density along the beam path, which influence the range of particle beams. Furthermore, in case of pencil beam scanning, there is an interference between the target movement and dynamic beam delivery. This review presents the strategies for tumour motion monitoring and moving target irradiation in the context of hadron therapy. Methods enabling the direct determination of tumour position (fluoroscopic imaging of implanted radio-opaque fiducial markers, electromagnetic detection of inserted transponders and ultrasonic tumour localization systems) are presented. Attention is also drawn to the techniques which use external surrogate motion for an indirect estimation of target displacement during irradiation. The role of respiratory-correlated CT [four-dimensional CT (4DCT)] in the determination of motion pattern prior to the particle treatment is also considered. An essential part of the article is the review of the main approaches to moving target irradiation in hadron therapy: gating, rescanning (repainting), gated rescanning and tumour tracking. The advantages, drawbacks and development trends of these methods are discussed. The new accelerators, called "cyclinacs", are presented, because their application to particle therapy will allow making a breakthrough in the 4D spot scanning treatment of moving organs.