Texture is a fundamental visual cue in computer vision which provides useful information about image regions. Dynamic Texture (DT) extends the analysis of texture to sequences of moving scenes. Classic approaches to texture and DT analysis are based on shallow hand-crafted descriptors including local binary patterns and filter banks. Deep learning and in particular Convolutional Neural Networks (CNNs) have significantly contributed to the field of computer vision in the last decade. These biologically inspired networks trained with powerful algorithms have largely improved the state of the art in various tasks such as digit, object and face recognition. This thesis explores the use of CNNs in texture and DT analysis, replacing classic hand-crafted filters by deep trainable filters. An introduction to deep learning is provided in the thesis as well as a thorough review of texture and DT analysis methods. While CNNs present interesting features for the analysis of textures such as a dense extraction of filter responses trained end to end, the deepest layers used in the decision rules commonly learn to detect large shapes and image layout instead of local texture patterns. A CNN architecture is therefore adapted to textures by using an orderless pooling of intermediate layers to discard the overall shape analysis, resulting in a reduced computational cost and improved accuracy. An application to biomedical texture images is proposed in which large tissue images are tiled and combined in a recognition scheme. An approach is also proposed for DT recognition using the developed CNNs on three orthogonal planes to combine spatial and temporal analysis. Finally, a fully convolutional network is adapted to texture segmentation based on the same idea of discarding the overall shape and by combining local shallow features with larger and deeper features.