Signal processing over graphs refers to a suite of methods that denoise, learn, and detect signals over graphs from noisy measurements at the vertices. This flexible framework has applications to sensor networks, neuroscience, genomics, social network and internet network analysis. We will give a brief overview of several algorithms for denoising signals over graphs including graph kernels, trend filtering over graphs, and graph wavelets. We will look at three applications of this methodology. First, we will use the fused lasso over graphs to adaptively segment road networks with terrorism event data. Second, we will explore trends in citation networks with dynamic network trend filtering, which will help us discover hot-spots of activity in an evolving corpus of publications. Third, we will use graph signal processing to estimate graphons---a flexible network model---in a locally adaptive fashion.