Reliable out-of-distribution (OOD) detection is a fundamental step towards a safer implementation of modern machine learning (ML) systems under distribution shift. In this paper, we introduce Igeood, an effective method for detecting OOD samples. Igeood applies to any pre-trained neural network, does not require OOD samples or assumptions on the OOD data, and works under different degrees of access to the ML model. By building on the geodesic (Fisher-Rao) distance between the underlying data distributions, our discriminator combines confidence scores from the logits outputs and the learned features of a deep neural network. Empirically, we show that Igeood outperforms competing state-of-the-art methods on a variety of networks architectures and datasets, e.g., by increasing up to 8.5% the average TNR at TPR-95% across six different models and nine different OOD datasets.