With an explosive development of the Internet-of-Things (IoT), the number of devices or machines (called users) connected to a base station is envisioned to reach tens of billions in the near future. In order to accommodate such a massive connectivity, random access schemes are deemed as an efficient method. While the random access schemes can reduce the signalling overhead, they result in the unknown user activity and the inevitable interference from contending users. This seriously hinders the application of random access schemes in many practical systems. Therefore, this thesis is dedicated to studying methods to improve the efficiency of random access schemes and to facilitate their deployment in machine-type communications (MTC). First, a joint user activity identification and channel estimation scheme is designed for grant-free random access systems. We propose a decentralized transmission control and design a compressed sensing (CS)-based user identification and channel estimation scheme. We analyze the packet delay and throughput of the proposed scheme. We also optimize the transmission control scheme to maximize the system throughput. Second, a random access scheme, i.e., the coded slotted ALOHA (CSA) scheme, is designed for erasure channels to improve the system throughput. By deriving the extrinsic information transfer (EXIT) functions and optimizing their convergence behavior, we design the code probability distributions for CSA schemes with repetition codes and maximum distance separable (MDS) codes to maximize the expected traffic load, under packet erasure channels and slot erasure channels. We derive the asymptotic throughput of the CSA schemes over the erasure channels for an infinite frame length, which is verified to well approximate the throughput for a practical frame length. Third, an efficient data decoding algorithm for the CSA scheme is proposed to further improve the system efficiency. We present a low-complexity physical-layer network coding (PNC) method to obtain linear combinations of collided packets, and design an enhanced message-level successive interference cancellation (SIC) algorithm to wisely exploit the linear combinations of collided packets. We propose an analytical framework and derive the system throughput for the proposed scheme. The CSA scheme is further optimized to maximize the system throughput and energy efficiency.