The new 5G’s wireless networks have started to be deployed all around the world. With them, a large spectrum of services are about to emerge, resulting in new stringent requirements so that 5G targets performances exceed that of 4G by a factor of 10. The services are enhanced mobile broadband (eMBB), ultra reliable and low-latency communication (uRLLC) and massive machine-type communication (mMTC) where each of which has required the ongoing development of key new technologies. Many of these technologies will also play an important role in the emergence of 6G. In this thesis, the focus is on grant-free RA (GFRA) as an enabler of uRLLC and mMTC. GFRA is a new protocol introduced in 5G new radio (5G-NR) for reducing the data overhead of the random access (RA) procedure. This results in a significant reduction in the latencies of the user equipments (UEs) access to a connected medium via an access point (AP). Achieving efficient GFRA is of key importance for many 5G applications, e.g. for large scale internet of things (IoT) wireless networks. The study of new non-orthogonal multiple access (NOMA) signal processing techniques is then considered. Using tools from the theory of Bayesian compressed sensing (CS), algorithms within the family of approximate message passing (AMP) are developed to address the joint active user detection and channel estimation (AUDaCE) problem. It is crucial to properly identify transmitting UEs and guarantee that an AP can reliably transmit back data to the detected UEs. In contrast to existing work on this topic, the AUDaCE is studied for wireless networks where the activity of the UEs is correlated. To this end, two activity models are introduced. The first one models the activity of the UEs in the network with group-homogeneous activity (GHomA). The second model accounts for more general dependence structure via group-heterogeneous activity (GHetA). Approximate message passing algorithms within the hybrid GAMP (HGAMP) framework are developed for each of the models. With the aid of latent variables associated to each group for modeling the activity probabilities of the UEs, the GHomA-HGAMP algorithm can perform AUDaCE for GFRA leveraging such a group homogeneity. When the activity is heterogenous, it is possible to develop GHetA-HGAMP using the copula theory. Extensive numerical studies are performed, which highlight significant performance improvements of GHomA-HGAMP and GHetA-HGAMP over existing algorithms which do not properly account for correlated activity. In particular, the channel estimation and active user detection capability are enhanced in many scenarios with up to a 4dB improvement with twice less user errors. As a whole, this thesis provides a systematic approach to AUDaCE for wireless networks with correlated activities using Bayesian CS.