The Sixth Generation and Beyond communication systems are expected to enable communications of a massive number of machine-type devices. The traffic generated by some of these devices will significantly deviate from those in conventional communication scenarios. For instance, for applications where a massive number of cheap sensors communicate with a base station (BS), the devices will only be sporadically active and there will be no coordination among them or with the BS. For such systems requiring massive random access solutions, a new paradigm called unsourced random access (URA) has recently been proposed. In URA, all the users employ the same codebook and there is no user identity. The destination is only interested in the list of messages being sent from the set of active users. While there are many interesting URA schemes developed in the recent literature, many significant challenges remain, in particular in designing low-complexity and energy-efficient solutions. With the motivation of addressing the current challenges in URA, we develop practical solutions for several scenarios. First, we propose and study URA over frequency-selective channels via orthogonal frequency division multiplexing to mitigate the fading effects. The decoder employs a joint activity detection and channel estimation algorithm coupled with treating interference as noise and successive interference cancellation (SIC). Our results show that the pro-posed scheme offers competitive performance with grant-based frequency division multiple-access while the performance loss due to the estimated channel state information is limited. We then examine the scenario for which the receiver is equipped with a massive number of antennas and develop a simple yet energy-efficient solution by dividing the transmission frame into slots where each active user utilizes a non-orthogonal pilot sequence followed by its polar encoded codeword. At the receiver, we first detect the transmitted pilot sequences by a generalized orthogonal matching pursuit algorithm and utilize a linear minimum mean square error (LMMSE) solution to estimate the channel vectors. We then perform iterative decoding based on maximal ratio combining and single-user decoding followed by SIC. Numerical examples and analysis results demonstrate that the proposed scheme either outperforms the existing approaches in the lit-erature or has a competitive performance with lower complexity. We then adapt our solution to the scenarios with residual hardware impairments (HWIs) at the BS and the user equipment sides by developing a hardware-impairment aware LMMSE solution for channel estimation using the HWI statistics and observe that the newly proposed solution improves the energy efficiency and increases the number of supported active users. Finally, we study on-off division multiple access in the context of URA where each active user utilizes a small fraction of the transmission frame and show that the new approach is superior to the existing ones in terms of performance or complexity.