This paper presents a method to classify flow regime and vapor quality in vertical two-phase (vapor-liquid) flow, using a video of the flow as the input; this represents the first high-performing and entirely camera image-based method for the classification of a vertical flow regime (which is effective across a wide range of regimes) and the first image-based tool for estimating vapor quality. The approach makes use of computer vision techniques and deep learning to train a convolutional neural network (CNN), which is used for individual frame classification and image feature extraction, and a deep long short-term memory (LSTM) network, used to capture temporal information present in a sequence of image feature sets and to make a final vapor quality or flow regime classification. This novel architecture for two-phase flow studies achieves accurate flow regime and vapor quality classifications in a practical application to two-phase CO2 flow in vertical tubes, based on offline data and an online prototype implementation, developed as a proof of concept for the use of these models within a feedback control loop. The use of automatically selected image features, produced by a CNN architecture in three distinct tasks comprising flow-image classification, flow-regime classification, and vapor quality prediction, confirms that these features are robust and useful, and offer a viable alternative to manually extracting image features for image-based flow studies. The successful application of the LSTM network reveals the significance of temporal information for image-based studies of two-phase flow.