Heart disease is a leading cause of death in the United States and abroad. Research interests arise in understanding the nature of the dynamics of the heart and seeking methods to control and suppress arrhythmias. Simulation of the heart electrical activity is a useful approach to study the heart because it yields some quantities of interest that cannot practically be obtained in any other way. However, the complexity of the human heart leads to complicated mathematical models, and consequently, modeling arrhythmias of a whole heart with computers is extremely data intensive and computational challenging. In this dissertation, we introduce an analog VLSI design that simulates cardiac electrical activities. The selected cardiac model is based on the Beeler-Reuter equations and the continuous core-conductor model. The Beeler-Reuter equations formulate the membrane ionic kinetics of ventricular cells, and the core-conductor model describes the electrical signal conduction on cardiac tissues. We discuss the design flows of mapping equations into circuits and present a set of circuit blocks of basic mathematical function units. The transistor circuits for realizing the ionic model of a single cell is introduced, and capacitors are used to calculate time directives. A method of shifting the initial conditions of differential equations to zero is discussed for saving the circuit which sets up the initial voltages of the capacitors. We also introduce a method of implementing reaction-diffusion systems using non-linear RC networks, and present the circuit which simulates the reaction-diffusion process, i.e. the electrical propagation, of the heart. Error analysis is carried out for the circuit-realized Beeler-Reuter model by comparing the simulated functions with the equation calculated values. The PSpice simulation results show that the circuit created action potential is satisfactory. The important reentry phenomena, the primary mechanism underlying fibrillation, is presented, and an anatomical reentry in the 1-dimensional model and a functional reentry (spiral wave) in the 2-dimensional model are successfully simulated in circuits. The presented methods of implementing equations with analog VLSI circuit contribute to the fundamentals for a novel technique of obtaining numerical solutions and potential fast application-specified analog computational devices if the circuits are fabricated on chips. Unlike computing with digital computers, which is mainly a serial process and needs to discretize the space and the time domain for finding numerical solutions of the discretization points one by one, computation with analog VLSI relies on the physics of the electrical devices and takes advantage of the integration properties of capacitors and, hence, computing in analog circuit hardware is a parallel process and can be real-time, that is, the calculation time is the time simulated by equations.