Convolutional Neural Networks (CNNs) constitute a class of Deep Learning models which have been used in the recent past to resolve many problems in computer vision, in particular optical flow estimation. Measuring displacement and strain fields can be regarded as a particular case of this problem. However, it seems that CNNs have never been used so far to perform such measurements. This work is aimed at implementing a CNN able to retrieve displacement and strain fields from pairs of reference and deformed images of a flat speckled surface, as Digital Image Correlation (DIC) does. This paper explains how a CNN called StrainNet can be developed to reach this goal, and how specific ground truth datasets are elaborated to train this CNN. The main result is that StrainNet successfully performs such measurements, and that it achieves competing results in terms of metrological performance and computing time. The conclusion is that CNNs like StrainNet offer a viable alternative to DIC, especially for real-time applications.