Chaos possesses many interesting properties, such as deterministic but random-like complex temporal behaviour, high sensitivity to initial conditions and system parameters, fractal structure, long-term unpredictability and so on. In the past two decades, these properties have been found to be useful for many engineering problems including cryptographic designs, digital communications, network behaviour modeling, to name a few. The increasing interests in utilizing chaotic dynamics in various applications have ignited tremendous demands for new chaos generators with complex dynamics but simple designs. This also motivates this piece of research work which targets for designing new methodologies to generate high-dimensional maps with nice characteristics, and exploring their possible applications, in particular, in cryptography. In this thesis, chaotic map is focused, and different methods for its generation have been explored and proposed. They include the chaotification of a linear discrete-time system, the coupling of multiple maps, the multidimensional generalization and the spatial extension of low dimensional chaotic map. Due to the differences of their formulations, the nature of the generated chaotic maps may not be the same and hence their characteristics are compared in details. The mixing property of these maps is of particular interests due to its prime importance for cryptographic designs. Recognizing that it is difficult to derive a mathematical proof on the mixing nature for such a high dimensional map, a statistical approach is designed and adopted. From the results, it shows that the method of spatial extension is more preferable and the corresponding maps possess the strongest mixing nature. With such nice and distinct properties on hands, the high-dimensional chaotic map generated by spatial extension has been suggested for two real-world cryptographic applications. Firstly, it is used to provide the authentication of data or message with a digested value based on a newly designed chaos-based hash function. The throughput of the proposed scheme is about 1.5 times of the existing method, MD5. In addition, a key feature can be easily embedded in the proposed scheme, and hence the computed hash value can be directly used as an authentication code or digital signature for the data without any further processing. The second application is a fast image encryption system under the framework of stream cipher. Based on the mixing nature of the high-dimensional chaotic map, random key sequences are generated and used to mask up images. A detailed statistical analysis on the proposed encryption scheme has been carried out and its effectiveness is confirmed.