This paper presents a robotic pick-and-place system that is capable of grasping and recognizing both known and novel objects in cluttered environments. The key new feature of the system is that it handles a wide range of object categories without needing any task-specific training data for novel objects. To achieve this, it first uses a category-agnostic affordance prediction algorithm to select and execute among four different grasping primitive behaviors. It then recognizes picked objects with a cross-domain image classification framework that matches observed images to product images. Since product images are readily available for a wide range of objects (e.g., from the web), the system works out-of-the-box for novel objects without requiring any additional training data. Exhaustive experimental results demonstrate that our multi-affordance grasping achieves high success rates for a wide variety of objects in clutter, and our recognition algorithm achieves high accuracy for both known and novel grasped objects. The approach was part of the MIT-Princeton Team system that took 1st place in the stowing task at the 2017 Amazon Robotics Challenge. All code, datasets, and pre-trained models are available online at http://arc.cs.princeton.edu.