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This work addresses the monitoring and state estimation of electrical grids, especially at the distribution level. For economic and technical reasons, grid monitoring cannot be implemented with a similarly high measurement density as in transmission grids. Two new monitoring methods, which are designed for low measurement density, are therefore presented for use in real-time grid operation. First, a heuristic monitoring method is presented, which does not require pseudo-measurements and estimates voltage magnitudes and line loadings. Second, a monitoring method based on artificial neural networks is presented. With appropriate training, the method can estimate grid variables, e.g., voltage magnitudes or line loadings, with high accuracy. The methods are tested on thousands of test scenarios using a comprehensive evaluation methodology. For measurement infrastructure planning, a concept is presented to determine suitable measurement locations for the use of one of the monitoring methods. After optimization, several possible measurement configurations are presented with their average and maximum errors and the projected capital expenditures.