Prosthetic mitral valves described herein can be deployed using a transcatheter mitral valve delivery system and technique to interface and anchor in cooperation with the anatomical structures of a native mitral valve. This document describes prosthetic heart valve designs that interface with native mitral valve structures to create a fluid seal, thereby minimizing mitral regurgitation and paravalvular leaks. This document also describes prosthetic heart valve designs and techniques to manage blood flow through the left ventricular outflow tract. In addition, this document describes prosthetic heart valve designs and techniques that reduce the risk of interference between the prosthetic valves and chordae tendineae.