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This thesis deals with the analysis and design of cryptographic hash functions that are fundamental components of many cryptographic applications such as digital signatures, authentication, key derivation, random number generation and many others. Due to this versality they are considered as the “Swiss army knives” of modern cryptology.A hash function is a one-way mathematical function that takes a message of arbitrary length as input and produces an output of fixed (smaller) length. In recent years, several of the approved cryptographic hash functions which are generally inspired by MD4 have been successfully attacked, and serious attacks have been published against the world-wide standard SHA-1. In response, the National Institute of Standards and Technology (NIST) has opened a public competition to develop a new cryptographic hash algorithm, SHA-3, to replace the older SHA-1 and SHA-2 hash functions.The first part of this thesis is focused on the analysis of the hash function JH, one of the finalists of this competition. We demonstrate attacks on JH showing that the algorithm is not as secure as claimed by its designer. We find a semi-free-start collision for the hash function and semi-free-start near- collisions for the compression function of reduced-round JH. Moreover, we present distinguishers for the full internal permutation.The second part of this thesis is focused on the design of hash functions. We propose a new family of sponge-based lightweight hash function called SPONGENT. We first explain the design strategy of SPONGENT and then we present its security analysis by applying the most important state-of-the-art methods of cryptanalysis and by investigating their complexity.