To satisfy the high data demands in future cellular networks, an ultra-densification approach is introduced to shrink the coverage of base station (BS) and improve the frequency reuse. The gain in capacity is expected but at the expense of increased interference, frequent handovers (HOs), increased HO failure (HOF) rates, increased HO delays, increase in ping pong rate, high energy consumption, increased overheads due to frequent HO, high packet losses and bad user experience mostly in high-speed user equipment (UE) scenarios. This paper presents the general concepts of radio access mobility in cellular networks with possible challenges and current research focus. In this article, we provide an overview of HO management in long-term evolution (LTE) and 5G new radio (NR) to highlight the main differences in basic HO scenarios. A detailed literature survey on radio access mobility in LTE, heterogeneous networks (HetNets) and NR is provided. In addition, this paper suggests HO management challenges and enhancing techniques with a discussion on the key points that need to be considered in formulating an efficient HO scheme.