Adapter modules were recently introduced as an efficient alternative to fine-tuning in NLP. Adapter tuning consists in freezing pretrained parameters of a model and injecting lightweight modules between layers, resulting in the addition of only a small number of task-specific trainable parameters. While adapter tuning was investigated for multilingual neural machine translation, this paper proposes a comprehensive analysis of adapters for multilingual speech translation (ST). Starting from different pre-trained models (a multilingual ST trained on parallel data or a multilingual BART (mBART) trained on non-parallel multilingual data), we show that adapters can be used to: (a) efficiently specialize ST to specific language pairs with a low extra cost in terms of parameters, and (b) transfer from an automatic speech recognition (ASR) task and an mBART pre-trained model to a multilingual ST task. Experiments show that adapter tuning offer competitive results to full fine-tuning, while being much more parameter-efficient.