The central theme of our work is the probabilistic and approximate design of embedded computing systems. This novel approach consists of two distinguishing aspects: (i) the design and implementation of embedded systems, using components which are susceptible to perturbations from various sources and (ii) a design methodology which consists of an exploration of a design space which characterizes the trade-off between quality of output and cost, to implement high performance and low energy embedded systems. In contrast with other work, our design methodology does not attempt to correct the errors introduced by components which are susceptible to perturbations, instead we design "good enough" systems. Our work has the potential to address challenges and impediments to Moore's law arising from material properties and manufacturing difficulties, which dictate that we shift from the current-day deterministic design paradigm to statistical and probabilistic designs of the future. In this paper, we provide a broad overview of our work on probabilistic and approximate design, present novel results in approximate arithmetic and its impact on digital signal processing algorithms, and sketch future directions for research.