Memristors are now becoming a prominent candidate to serve as the building blocks of non-von Neumann in-memory computing architectures. By mapping analog numerical matrices into memristor crossbar arrays, efficient multiply accumulate operations can be performed in a massively parallel fashion using the physics mechanisms of Ohm’s law and Kirchhoff’s law. In this brief review, we present the recent progress in two niche applications: neural network accelerators and numerical computing units, mainly focusing on the advances in hardware demonstrations. The former one is regarded as soft computing since it can tolerant some degree of the device and array imperfections. The acceleration of multiple layer perceptrons, convolutional neural networks, generative adversarial networks, and long short-term memory neural networks are described. The latter one is hard computing because the solving of numerical problems requires high-precision devices. Several breakthroughs in memristive equation solvers with improved computation accuracies are highlighted. Besides, other nonvolatile devices with the capability of analog computing are also briefly introduced. Finally, we conclude the review with discussions on the challenges and opportunities for future research toward realizing memristive analog computing machines.