Compared to the generations up to 4G, whose main focus was on broadband and coverage aspects, 5G has expanded the scope of wireless cellular systems towards embracing two new types of connectivity: massive machine-type communication (mMTC) and ultra-reliable low-latency communications (URLLC). This paper discusses the possible evolution of these two types of connectivity within the umbrella of 6G wireless systems. The paper consists of three parts. The first part deals with the connectivity for a massive number of devices. While mMTC research in 5G predominantly focuses on the problem of uncoordinated access in the uplink for a large number of devices, the traffic patterns in 6G may become more symmetric, leading to closed-loop massive connectivity. One of the drivers for this is distributed learning/inference. The second part of the paper discusses the evolution of wireless connectivity for critical services. While latency and reliability are tightly coupled in 5G, 6G will support a variety of safety critical control applications with different types of timing requirements, as evidenced by the emergence of metrics related to information freshness and information value. Additionally, ensuring ultra-high reliability for safety critical control applications requires modeling and estimation of the tail statistics of the wireless channel, queue length, and delay. The fulfillment of these stringent requirements calls for the development of novel AI-based techniques, incorporating optimization theory, explainable AI, generative AI and digital twins. The third part analyzes the coexistence of massive connectivity and critical services. We will consider scenarios in which a massive number of devices need to support traffic patterns of mixed criticality. This is followed by a discussion about the management of wireless resources shared by services with different criticality.