We introduce a new family of neural network models called Convolutional Dynamic Alignment Networks (CoDA Nets), which are performant classifiers with a high degree of inherent interpretability. Their core building blocks are Dynamic Alignment Units (DAUs), which are optimised to transform their inputs with dynamically computed weight vectors that align with task-relevant patterns. As a result, CoDA Nets model the classification prediction through a series of input-dependent linear transformations, allowing for linear decomposition of the output into individual input contributions. Given the alignment of the DAUs, the resulting contribution maps align with discriminative input patterns. These model-inherent decompositions are of high visual quality and outperform existing attribution methods under quantitative metrics. Further, CoDA Nets constitute performant classifiers, achieving on par results to ResNet and VGG models on e.g. CIFAR-10 and TinyImagenet. Lastly, CoDA Nets can be combined with conventional neural network models to yield powerful classifiers that more easily scale to complex datasets such as Imagenet whilst exhibiting an increased interpretable depth, i.e., the output can be explained well in terms of contributions from intermediate layers within the network