The frequent occurrence of transcription and DNA replication in cells results in many encounters, and thus conflicts, between the transcription and replication machineries. These conflicts constitute a major intrinsic source of genome instability, which is a hallmark of cancer cells. How the replication machinery progresses along a DNA molecule occupied by an RNA polymerase is an old question. Here we review recent data on the biological relevance of transcription-replication conflicts, and the factors and mechanisms that are involved in either preventing or resolving them, mainly in eukaryotes. On the basis of these data, we provide our current view of how transcription can generate obstacles to replication, including torsional stress and non-B DNA structures, and of the different cellular processes that have evolved to solve them.