The success of deep learning has enabled advances in multimodal tasks that require non-trivial fusion of multiple input domains. Although multimodal models have shown potential in many problems, their increased complexity makes them more vulnerable to attacks. A Backdoor (or Trojan) attack is a class of security vulnerability wherein an attacker embeds a malicious secret behavior into a network (e.g. targeted misclassification) that is activated when an attacker-specified trigger is added to an input. In this work, we show that multimodal networks are vulnerable to a novel type of attack that we refer to as Dual-Key Multimodal Backdoors. This attack exploits the complex fusion mechanisms used by state-of-the-art networks to embed backdoors that are both effective and stealthy. Instead of using a single trigger, the proposed attack embeds a trigger in each of the input modalities and activates the malicious behavior only when both the triggers are present. We present an extensive study of multimodal backdoors on the Visual Question Answering (VQA) task with multiple architectures and visual feature backbones. A major challenge in embedding backdoors in VQA models is that most models use visual features extracted from a fixed pretrained object detector. This is challenging for the attacker as the detector can distort or ignore the visual trigger entirely, which leads to models where backdoors are over-reliant on the language trigger. We tackle this problem by proposing a visual trigger optimization strategy designed for pretrained object detectors. Through this method, we create Dual-Key Backdoors with over a 98% attack success rate while only poisoning 1% of the training data. Finally, we release TrojVQA, a large collection of clean and trojan VQA models to enable research in defending against multimodal backdoors.