Mobile robot localization is one of the most fundamental problems in robotics. For robots assisting humans in tasks such as surveillance, search and rescue, and space exploration, accurate localization, that is, precisely estimating the robot's pose (position and orientation), is a prerequisite for autonomous operation. The system resources (processing and communication) for localization, however, are often limited, and their availability varies widely depending upon the application and the operating environment. Therefore, the objective of this work is to develop resource-aware estimators for robot localization, which optimally utilize all available resources in order to maximize estimation accuracy. In the first part of this thesis, we address the problem of robot localization under processing constraints, focusing on the key applications of single-robot Simultaneous Localization and Mapping (SLAM) and multi-robot Cooperative Localization (CL). For SLAM, we propose two resource-aware approaches, the approximate Minimum Mean Squared Error (MMSE) estimator-based Power-SLAM algorithm and the approximate batch Maximum A Posterior (MAP) estimator-based Constrained Keyframe-based Localization and Mapping (C-KLAM). When approximations are inevitable due to processing constraints, both approaches aim to minimize the information loss while generating consistent estimates. For CL, we exploit the sparse structure of the batch MAP estimator to develop a resource-aware, fully-distributed multi-robot localization algorithm, that harnesses the processing, storage, and communication resources of the entire team, to obtain substantial speed-up. The second part of this thesis focuses on CL under communication constraints, in particular, asynchronous communication and bandwidth constraints. Due to limited communication range or the presence of obstacles, robots communicate asynchronously, that is, they can only interact with different sub-teams over time and exchange information intermittently. For this scenario, we develop a family of resource-aware information exchange rules for the robots, in order to ensure optimal and consistent localization performance. Lastly, this thesis investigates the problem of decentralized estimation under stringent communication bandwidth constraints. Here, robots can communicate only a severely quantized version (few or only one bit), of their real-valued sensor measurements, to the team. Existing estimation frameworks, however, are designed to process either real-valued or quantized measurements. To overcome this drawback, we propose a paradigm shift in estimation methodology by focusing on the design and performance evaluation of the first-ever, resource-aware, hybrid estimators. The proposed hybrid estimators are able to process both locally-available real-valued information, along with the quantized information received from the team, in order to maximize localization accuracy. Finally, we note that mobile robot applications are no longer limited to specialized and expensive robots. Commonly-available hand-held devices such as cell phones, PDAs, and even cars, are equipped with processing, sensing, and networking capabilities. Therefore, when coupled with the proposed innovative, scalable, and resource-aware algorithms, these ubiquitous mobile devices can lead to a proliferation of novel location-based services.