Inverse problems are concerned with determining causes by knowledge of consequences. In its abstract formulation, an inverse problem can be given as the following operator equation: \(\displaystyle \mathcal {F}(\mathrm {x})=\mathrm {y},\quad x\in \mathbb {X},\ \mathrm {y}=\mathbb {Y}, \) where \(\mathbb {X}\) and \(\mathbb {Y}\) , respectively, denote the sets of target objects and observables, and \(\mathcal {F}\) denote a mapping from \(\mathbb {X}\) to \(\mathbb {Y}\) . The inverse problem is concerned with determining \(\mathrm {x}\) or missing information in \(\mathcal {F}\) by knowledge of \(\mathrm {y}\) . In this chapter, we discuss several fundamental issues including modeling, unique identifiability, stability estimates, and reconstruction algorithms for the general study of inverse problems. We also elaborate the discussion on the specific EIT (Electrical Impedance Tomography) problem. Finally, we review some basic concepts on integro-differential operators as well as the associated inverse problems.