Speech quality in conventional telephony is degraded, since only a fraction of the original acoustic speech bandwidth is transmitted. Artificial speech bandwidth extension (ABE) is a means to recover missing frequency components to increase speech quality and intelligibility. Whenever larger acoustic speech bandwidths are not available, ABE can serve as fallback solution, since it can be used independently of the communication system. In this work, ABE approaches have been developed to extend the acoustic bandwidth of speech signals towards higher and lower frequencies in order to increase the perceived speech quality. For the extension of higher frequencies, deep neural networks (DNNs) are employed to establish a link between the available bandwidth and missing high-frequency regions, whereas the extension towards lower frequencies is based on a robust signal model, considering the properties of low-frequency components in speech signals. In subjective listening tests, all of the developed ABE solutions for an extension towards higher and lower frequencies were found to improve the speech quality. Additionally, speech intelligibility and quality could be increased for persons compensating their profound deafness by a cochlear implant using a DNN-based ABE approach. Furthermore, an instrumental measure for predicting the speech quality of ABE-processed speech signals has been developed, since existing measures are not well suited for this task. Good generalization capabilities of the developed instrumental measure to accurately predict the speech quality of ABE-processed speech were proven in scenarios of unknown speech material, unknown languages, and, most importantly, unknown ABE solutions.