Conventional transport processes are driven by biases, e.g., diffusion is driven by concentration biases, heat flow is driven by temperature biases. Anomalous transport, although rarely seen in our daily lives, is also important. For instance, intracellular transport cannot only rely on diffusion, rather, cargoes are transported through directed motion of molecular motors along microtubules. Compared with the anomalous transport of mass, the transport of energy and momentum are much less well studied. In this thesis, we describe these two types of anomalous transport in a few classes of chiral active systems. The first class is active gyroscopic networks, which we construct by subjecting spring-mass networks to unbiased nonequilibrium fluctuations and Lorentz forces. We numerically demonstrate the emergence of energy rectification, in other words energy transport in the absence of temperature biases, between nodes for unmodulated networks, and between nodes and baths in time-modulated networks. We develop analytical diagrammatic theories that allow us to understand and control the rectification in arbitrary complex networks in terms of local properties. The second class is chiral active liquids. Previous studies have shown the existence of anomalous transport coefficients such as the so-called odd viscosity in these systems. Extending the Irving-Kirkwood theory for conventional fluids, we develop a theory for anomalous transport in chiral active liquids. We show how the transport coefficients can be connected to local molecular properties, specifically the averaged intermolecular forces and distortions of pair correlation functions. Taken together, we have contributed to the topic of anomalous transport of energy and momentum. Our analytic theories have provided frameworks to understand complex transport processes in terms of relatively simple local properties.