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Given the fact that a profound investigation of cellular automata (CAs) has only become possible following the deployment of modern computers, as opposed partial differential equations or maps that which have been studied extensively throughout the last centuries, the intriguing CA dynamics is often still not completely understood. In addition to the computational challenge a profound analysis of CAs poses, their discrete nature hinders the direct use of many mathematical techniques as these are often pervaded by the existence of the continuum. During this talk it will be demonstrated how one can nonetheless analyze CA dynamics within a rigorous framework that relies on Boolean derivatives and non-directional Lyapunov exponents. Besides, it will be shown how these tools can be used to gain insight into the interference between a CA's dynamics, on the one hand, and its topology, synchronism, and so on, on the other hand.