Twitter has transformed into one of the main platforms for citizen engagement today. However, even though previous studies have focused on opinions about vaccines in general or about specific vaccines, opinions towards COVID-19 vaccines on Twitter have not been researched to date. The objective of this research is, by using social network analysis and language processing tools, to examine the degree to which the opinions and interactions present on Twitter are favorable or unfavorable towards the main COVID-19 vaccines. In addition, the relevance of each of the vaccines is studied, as well as their level of controversy. Likewise, the present study investigates, for the first time, the conversation from different perspectives including the content and also the participants, by analyzing in detail the verified accounts and using tools for the detection of bots. In global terms, the results from verified accounts show a moderate favorability towards the COVID-19 vaccines, the most accepted being those of Oxford-AstraZeneca, Pfizer, Moderna, and Sputnik V. On the other hand, the vaccine that attracts the most attention is the Russian Sputnik V, which is also the most controversial, behind those developed in China. Finally, verified users are shown to be relevant agents in the conversation due to their greater capacity for dissemination and reach, while the presence of bots is practically non-existent.