The severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) not only caused the COVID-19 pandemic but also had a major impact on farmed mink production in several European countries. In Denmark, the entire population of farmed mink (over 15 million animals) was culled in late 2020. During the period of June to November 2020, mink on 290 farms (out of about 1100 in the country) were shown to be infected with SARS-CoV-2. Genome sequencing identified changes in the virus within the mink and it is estimated that about 4000 people in Denmark became infected with these mink virus variants. However, the routes of transmission of the virus to, and from, the mink have been unclear. Phylogenetic analysis revealed the generation of multiple clusters of the virus within the mink. Detailed analysis of changes in the virus during replication in mink and, in parallel, in the human population in Denmark, during the same time period, has been performed here. The majority of cases in mink involved variants with the Y453F substitution and the H69/V70 deletion within the Spike (S) protein; these changes emerged early in the outbreak. However, further introductions of the virus, by variants lacking these changes, from the human population into mink also occurred. Based on phylogenetic analysis of viral genome data, we estimate, using a conservative approach, that about 17 separate examples of mink to human transmission occurred in Denmark but up to 59 such events (90% credible interval: (39–77)) were identified using parsimony to count cross-species jumps on transmission trees inferred using Bayesian methods. Using the latter approach, 136 jumps (90% credible interval: (117–164)) from humans to mink were found, which may underlie the farm-to-farm spread. Thus, transmission of SARS-CoV-2 from humans to mink, mink to mink, from mink to humans and between humans were all observed.