Electrical impedance spectroscopy has been used extensively for sensing and biosensing due to the multiple electrical properties that can be interrogated through varying the frequency of the electrical excitation. In this paper, we review the basic concepts and key issues for applying impedance spectroscopy in sensing and biosensing, with emphasis on the development of precise, low-cost and portable instruments. An impedance spectroscopy system can be divided into three parts: the signal processing unit, the sensing unit and the data analysis unit. Herein, we focus on the signal processing unit, responsible for generating the excitation signal and performing the impedance readout. Special attention is given to small and low-cost signal processing circuits, which are essential for portability and point-of-care diagnosis. We also elaborate upon the various methods to fabricate the sensing units, including the choice of nanomaterials and biomolecules in controlled molecular architectures. From an instrumentation perspective, we discuss possible sources of interference in the measurement protocols. When impedance spectroscopy measurements are performed with arrays of sensing units, as with electronic tongues, large amounts of data are generated. This has motivated an increasing use of statistical and computational methods for data analysis. We present some of these methods, with examples of information visualization and machine learning techniques, which have been employed in analyzing impedance spectroscopy data in recent years.