I will present an introduction to functions of bounded variation, 1-laplacian type equations, and least gradient problems. I shall discuss the applications of such problems to the inverse problem of recovering the electrical conductivity outside some perfectly conducting or insulating inclusions from the interior measurement of the magnitude of one current density field. We will also consider this problem on electrical networks, i.e. the problem of determining the conductivity matrix of an electrical network from the induced current along the edges. I shall talk about the inverse problem of determining transition probabilities for random walks on graphs from the knowledge of the net number of times a random walker passes along the edges of the graph. The above problems hold potential for direct impact in medical imaging, analysis of computer and social networks, cryptography, and biology. At the same time, they lead to beautiful and challenging problems in analysis, geometric measure theory, weighted least gradient problems, theory of minimal surfaces, and numerical analysis.