Cognification – the act of transforming ordinary objects or processes into their intelligent counterparts through Data Science and Artificial Intelligence – is a disruptive technology that has been revolutionizing disparate fields ranging from corporate law to medical diagnosis. Easy access to massive data sets, data analytics tools and High-Performance Computing (HPC) have been fueling this revolution. In many ways, cognification is similar to the electrification revolution that took place more than a century ago when electricity became a ubiquitous commodity that could be accessed with ease from anywhere in order to transform mechanical processes into their electrical counterparts. In this paper, we consider two particular forms of distributed computing – Data Centers and HPC systems – and argue that they are ripe for cognification of their entire ecosystem, ranging from top-level applications down to low-level resource and power management services. We present our vision for what "Cognified Distributed Computing" might look like and outline some of the challenges that need to be addressed and new technologies that need to be developed in order to make it a reality. In particular, we examine the role cognification can play in tackling power consumption, resiliency and management problems in these systems. We describe intelligent software-based solutions to these problems powered by on-line predictive models built from streamed real-time data. While we cast the problem and our solutions in the context of large Data Centers and HPC systems, we believe our approach to be applicable to distributed computing in general. We believe that the traditional systems research agenda has much to gain by crossing discipline boundaries to include ideas and techniques from Data Science, Machine Learning and Artificial Intelligence.