Channel coding is at the heart of any communications systems and plays a key role in meeting the requirements of current and future wireless standards. The services defined by the 3rd Generation Partnership Project (3GPP) for fifth generation (5G) new radio, in particular ultra-reliable low-latency communication (URLLC) and massive machine type communication (mMTC), have very stringent requirements in terms of latency and reliability. Short packet communication is proved to be one of the solutions to achieve the latency requirement of future wireless communication systems due to shorter transmission-time interval. Therefore, designing robust and efficient short channel codes for URLLC and mMTC is of critical importance to design sustainable and efficient mobile communication systems. Traditional channel coding methods are not able to fulfil the requirements for Beyond 5G (B5G) or 6th generation (6G) mobile standards, which have a strong emphasis on bit-level granularity and flexibility. Rateless codes have been considered as viable candidates for short packet communication in URLLC and mMTC due to their rate-adaptive nature. Furthermore, the fact that rateless codes do not rely on channel state information (CSI) makes them stand out from traditional channel codes, which is an important characteristic to reduce transmission overhead. Analog fountain code (AFC) is a newly introduced rateless code, which has linear complexity in terms of encoding and decoding, and a capacity-approaching performance for a wide range of signal-to-noise ratios (SNRs). The code structure is simple; that is, the modulated information symbols are directly generated from information symbols in a linear manner. In this thesis, I will take a step forward to provide a comprehensive analysis and design of AFC for short packet communications. I first propose a density evolution (DE)-based framework, which tracks the evolution of the probability density function of the messages exchanged between AFC’s variable and check nodes in the belief propagation (BP) decoder. Based on the proposed DE framework, I formulate an optimisation problem to find the optimal weight set for AFC in order to minimise the bit error rate at a given SNR. Simulation results show the superiority of the AFC code design with optimised weight set compared to existing AFC designs in the literature. Next, I focus on the design of AFC in a short block length regime. In order to guarantee the performance of AFC in these conditions, a proper precoder is required. Therefore, the optimised weight set obtained from the proposed DE framework can be directly applied in the short block length regime with the aid of a precoder with powerful error correcting capability. Specifically, I use Bose, Chaudhuri, and Hocquenghem (BCH) codes with an ordered statistics decoder (OSD) and low-density parity check (LDPC) codes with the BP decoder. Simulation results show that low rate precoders offer better reliability across a wide range of SNRs compared to high-rate precoders. Additionally, the precoded AFC performs close to the normal approximation benchmark in the short block length regime over a wide range of SNRs. I also discuss the complexity of the AFC decoder and propose a threshold-based decoder to reduce it. Finally, I focus on the analysis and design of AFC in a multiple access channel. I present two encoding schemes, i.e., superposition coding and joint encoding, and two decoding schemes, i.e., successive interference cancellation and joint decoding, for multiple access AFC (MA-AFC). The process of joint encoding and the updating rules for joint decoding are explained in detail. I propose different combinations of encoding and decoding schemes for MA-AFC and evaluate their performance in terms of block error rate (BLER) to determine which combination has the best performance. Next, I propose a DE-based framework to track the evolution of messages exchanged between check nodes and variable nodes of MA-AFC under the joint decoding scheme. With the proposed DE framework, I formulate optimisation problems to find the optimal AFC code parameters, specifically the weight-set, which minimise the bit error rate at a given SNR. Simulation results show that the optimised AFC code outperforms the existing AFC code design in multiple access scenarios.