We present a review of the different aspects associated with the interaction of successive coronal mass ejections (CMEs) in the corona and inner heliosphere, focusing on the initiation of series of CMEs, their interaction in the heliosphere, the particle acceleration associated with successive CMEs, and the effect of compound events on Earth's magnetosphere. The two main mechanisms resulting in the eruption of series of CMEs are sympathetic eruptions, when one eruption triggers another, and homologous eruptions, when a series of similar eruptions originates from one active region. CME-CME interaction may also be associated with two unrelated eruptions. The interaction of successive CMEs has been observed remotely in coronagraphs (with the Large Angle and Spectrometric Coronagraph Experiment -LASCO-since the early 2000s) and heliospheric imagers (since the late 2000s), and inferred from in situ measurements, starting with early measurements in the 1970s. The interaction of two or more CMEs is associated with complex phenomena, including magnetic reconnection, momentum exchange, the propagation of a fast magnetosonic shock through a magnetic ejecta, and changes in the CME expansion. The presence of a preceding CME a few hours before a fast eruption has been found to be connected with higher fluxes of solar energetic particles (SEPs), while CME-CME interaction occurring in the corona is often associated with unusual radio bursts, indicating electron acceleration. Higher suprathermal population, enhanced turbulence and wave activity, stronger shocks, and shock-shock or shock-CME interaction have been proposed as potential physical mechanisms to explain the observed associated SEP events. When measured in situ, CME-CME interaction may be associated with relatively well organized multiple-magnetic cloud events, instances of shocks propagating through a previous magnetic ejecta or more complex ejecta, when the characteristics of the individual eruptions cannot be easily distinguished. CME-CME interaction is associated with some of the most intense recorded geomagnetic storms. The compression of a CME by another and the propagation of a shock inside a magnetic ejecta can lead to extreme values of the southward magnetic field component, sometimes associated with large values of the dynamic pressure. This can result in intense geomagnetic storms, but also trigger substorms and large earthward motions of the magnetopause, potentially associated with changes in the outer radiation belts. Future in situ measurements in the inner heliosphere by Solar Probe+ and Solar Orbiter may shed light on the evolution of CMEs as they interact, by providing opportunities for conjunction and evolutionary studies.