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It is widely recognised that future smart grids will heavily rely upon intelligent communication and signal processing as enabling technologies for their operation. Traditional tools for power system analysis, which have been built from a circuit theory perspective, are a good match for balanced system conditions. However, the unprecedented changes that are imposed by smart grid requirements, are pushing the limits of these old paradigms. To this end, we provide new signal processing perspectives to address some fundamental operations in power systems such as frequency estimation, regulation and fault detection. Firstly, motivated by our finding that any excursion from nominal power system conditions results in a degree of non-circularity in the measured variables, we cast the frequency estimation problem into a distributed estimation framework for noncircular complex random variables. Next, we derive the required next generation widely linear, frequency estimators which incorporate the so-called augmented data statistics and cater for the noncircularity and a widely linear nature of system functions. Uniquely, we also show that by virtue of augmented complex statistics, it is possible to treat frequency tracking and fault detection in a unified way. To address the ever shortening time-scales in future frequency regulation tasks, the developed distributed widely linear frequency estimators are equipped with the ability to compensate for the fewer available temporal voltage data by exploiting spatial diversity in wide area measurements. This contribution is further supported by new physically meaningful theoretical results on the statistical behavior of distributed adaptive filters. Our approach avoids the current restrictive assumptions routinely employed to simplify the analysis by making use of the collaborative learning strategies of distributed agents. The efficacy of the proposed distributed frequency estimators over standard strictly linear and stand-alone algorithms is illustrated in case studies over synthetic and real-world three-phase measurements. An overarching theme in this thesis is the elucidation of underlying commonalities between different methodologies employed in classical power engineering and signal processing. By revisiting fundamental power system ideas within the framework of augmented complex statistics, we provide a physically meaningful signal processing perspective of three-phase transforms and reveal their intimate connections with spatial discrete Fourier transform (DFT), optimal dimensionality reduction and frequency demodulation techniques. Moreover, under the widely linear framework, we also show that the two most widely used frequency estimators in the power grid are in fact special cases of frequency demodulation techniques. Finally, revisiting classic estimation problems in power engineering through the lens of non-circular complex estimation has made it possible to develop a new self-stabilising adaptive three-phase transformation which enables algorithms designed for balanced operating conditions to be straightforwardly implemented in a variety of real-world unbalanced operating conditions. This thesis therefore aims to help bridge the gap between signal processing and power communities by providing power system designers with advanced estimation algorithms and modern physically meaningful interpretations of key power engineering paradigms in order to match the dynamic and decentralised nature of the smart grid.