Developing systems that can synthesize natural and life-like motions for simulated characters has long been a focus for computer animation. But in order for these systems to be useful for downstream applications, they need not only produce high-quality motions, but must also provide an accessible and versatile interface through which users can direct a character's behaviors. Natural language provides a simple-to-use and expressive medium for specifying a user's intent. Recent breakthroughs in natural language processing (NLP) have demonstrated effective use of language-based interfaces for applications such as image generation and program synthesis. In this work, we present PADL, which leverages recent innovations in NLP in order to take steps towards developing language-directed controllers for physics-based character animation. PADL allows users to issue natural language commands for specifying both high-level tasks and low-level skills that a character should perform. We present an adversarial imitation learning approach for training policies to map high-level language commands to low-level controls that enable a character to perform the desired task and skill specified by a user's commands. Furthermore, we propose a multi-task aggregation method that leverages a language-based multiple-choice question-answering approach to determine high-level task objectives from language commands. We show that our framework can be applied to effectively direct a simulated humanoid character to perform a diverse array of complex motor skills.