Edge computing (EC) can overcome several limitations of cloud computing. In the EC environment, a service provider can deploy its application instances on edge servers to serve users with low latency. Given a limited budget K for deploying applications in a particular geographical area, some approaches have been proposed to achieves various optimization objectives, e.g., to maximize the servers' coverage, to minimize the average network latency, etc. However, the robustness of the services collectively delivered by the service provider's applications deployed on the edge servers has not been considered at all. This is a critical issue, especially in the highly distributed, dynamic and volatile EC environment. We make the first attempt to tackle this challenge. Specifically, we formulate this Robustness-oriented Edge Application Deployment(READ) problem as a constrained optimization problem and prove its NP-hardness. Then, we provide an integer programming based approach READ-O for solving it precisely, and an approximation algorithm READ-A for efficiently finding near-optimal solutions to large-scale problems. READ-A's approximation ratio is not worse than K/2, which is constant regardless of the total number of edge servers. Evaluation of the widely-used real-world dataset against five representative approaches demonstrates that our approaches can solve the READ problem effectively and efficiently.