Context. Despite the importance of magnetic fields in massive stars, their origin is widely debated and still not well understood. Aims. With the mounting evidence for the importance of studying magnetic fields in interacting massive binary and multiple systems, it appears necessary to investigate the presence of magnetic fields in semi-detached systems with ongoing mass transfer, and in contact systems where mass is actively being exchanged. Methods. We present an analysis of 53 high-resolution HARPSpol spectropolarimetric observations of a sample of 14 massive binary and multiple systems using the least-squares deconvolution technique. The majority of the studied systems are classified as semi-detached or contact binaries. Results. Definite detections of the presence of a magnetic field are achieved in all studied systems apart from the rather faint system SV Cen, for which only a marginal detection was obtained. The fact that the presence of magnetic fields is detected in all but one of the studied systems strongly suggests that interaction between the system components plays a definite role in the generation of magnetic fields in massive stars. The measured mean longitudinal magnetic field strength for all targets is of the order of a few hundred Gauss to a few kiloGauss. The strongest longitudinal magnetic fields of 4─5 kG are discovered in the massive O-type triple system MY Ser in both components of the contact binary. kiloGauss-order magnetic fields are also detected in two other systems, V1294 Sco and V606 Cen. It is possible that there is an implication of some system characteristics, such as multiplicity, the mass ratio between the components, and a large fillout factor, on the measured magnetic field strength. Our results for the magnetic field measurements in interacting binaries present the first assessment of the occurrence rate of magnetic fields in a representative sample of such systems.