Computing has a huge memory problem. The memory system, consisting of multiple technologies at different levels, is responsible for most of the energy consumption, performance bottlenecks, robustness problems, monetary cost, and hardware real estate of a modern computing system. All this becomes worse as modern and emerging applications become more data-intensive (as we readily witness in e.g., machine learning, genome analysis, graph processing, and data analytics), making the memory system an even larger bottleneck. In this paper, we discuss two major challenges that greatly affect computing system performance and efficiency: 1) memory technology & capacity scaling (at the lower device and circuit levels) and 2) system and application performance & energy scaling (at the higher levels of the computing stack). We demonstrate that both types of scaling have become extremely difficult, wasteful, and costly due to the dominant processor-centric design & execution paradigm of computers, which treats memory as a dumb and inactive component that cannot perform any computation. We show that moving to a memory-centric design & execution paradigm can solve the major challenges, while enabling multiple other potential benefits. In particular, we demonstrate that: 1) memory technology scaling problems (e.g., RowHammer, RowPress, Variable Read Disturbance, data retention, and other issues awaiting to be discovered) can be much more easily and efficiently handled by enabling memory to autonomously manage itself; 2) system and application performance & energy efficiency can, at the same time, be improved by orders of magnitude by enabling computation capability in memory chips and structures (i.e., processing in memory). We discuss adoption challenges against enabling memory-centric computing, and describe how we can get there step-by-step via an evolutionary path.