The dissertation provides polar coding techniques for a variety of source and channel models with applications to storage and communication networks. We first provide universal polar codes for asymmetric compound channels that avoid common randomness. A staircase alignment of polar blocks is considered in the code construction. An MDS code is used in each column achieving the universality and a scrambling technique is implemented for each column helping avoid common randomness. These compound asymmetric channels are used for modelling flash-memories, such as MLCs (multi-level cell flash memories), and TLCs (three-level cell flash memories) memories. Hence the proposed universal polar codes for asymmetric channels can be used for flash memory error correction. The costly noiseless channel model was used to model a flash memory device. Each of the voltage levels to which a flash memory cell can be programmed has an associated wear cost which reflects the damage caused to the cell by repeated programming to that level. Shaping codes that minimize the average cost per channel symbol for a specified rate and shaping codes that minimize the average cost per source symbol (i.e., the total cost) have been shown to reduce cell wear and increase the lifetime of the memory. Hence, we study polar shaping codes for costly noiseless channels minimizing total cost. We also study polar shaping codes for costly noisy channels for the design of efficient codes that combine wear reduction and error correction for use in a noisy flash memory device. A novel scheme based on polar codes is proposed to compress a uniform source when a side information correlated with the source is available at the receiver while the conditional distribution of the side information given the source is symmetric and unknown to the source. An adaptation of universal polar codes with an incorporation of the linear code duality between channel coding and Slepian-Wolf coding is used in the design of those codes. Optimal rate is achieved through the proposed codes for the source model. These codes can be used in a wireless sensor network where the measurements tracked at two different nodes are correlated and the correlation may not always be fixed due to environmental changes such as weather. The nodes communicate the information sensed or measured by them to a central location. Finally, we provide a capacity-achieving polar coding strategy on a multi-level 3-receiver broadcast channel in which the second receiver is degraded (stochastically) from the first receiver for the transmission of a public message intended for all the receivers and a private message intended for the first receiver. A chaining strategy, translating the ideas of superposition coding, rate-splitting and indirect coding into polar coding, is used in the construction. The codes designed for such a channel model and setting can be used for video and audio file transfer in a client-server network where the individual clients are a computer and two mobile phones. The two mobile phone clients just support audio application where the computer supports both audio and video applications.