This thesis is devoted to the study of the dynamics of spiral and scroll waves in a mathematical model for cardiac tissue. We study the effects of the presence of heterogeneities on electrical-wave dynamics. The heterogeneities in the medium occur because of the variation in the electrophysiological properties of the constituent myocytes in the tissue, or because of the presence of cells like fibroblasts and pathological myocytes that can trigger early afterdepolarizations (EADs). We study how these heterogeneities can lead to the formation of spiral and scroll waves and how they can affect the stability of the spiral and scroll waves in cardiac tissue. We also investigate the role of abnormal cells, which can trigger pathological excitations like EADs, on the formation of spiral and scroll waves, and how such cells can trigger premature electrical pulses like premature-ventricular-complexes (PVCs) in cardiac tissue. Earlier studies have examined the role of ionic heterogeneities on spiral-wave initiation and their effects on spiral-wave stability. However, none of these studies has calculated, in a controlled way, the effects of individual ion-channel conductances on spiral- and scroll-wave properties, such as the frequency of these waves, and the effects of the spatial gradients, in each ion-channel conductance, on their stability; we present these results in Chapter 2. Although many studies in the past have studied the effects of fibroblast coupling on wave-dynamics in cardiac tissue, a detailed study of spiral-wave dynamics in a medium with a well-defined, heterogeneous distribution of fibroblasts (e.g., with a gradient in the fibroblast density (GFD)) has not been performed; therefore, in Chapter 3 we present the effects of such GFD on spiral- and scroll-wave dynamics. Then, in Chapter 4, we present a systematic study of how a clump of fibroblasts can lead to spiral waves via high-frequency pacing. Some studies in the past have studied the role of early afterdepolarizations (EADs) in the formation of arrhythmias in cardiac tissue; we build on such studies and present a detailed study of the effects of EADs on the formation of spiral waves and their dynamics, in Chapter 5. Finally, in Chapter 6 we provide the results of our detailed investigation of the factors that assist the triggering of abnormal electrical pulses like premature ventricular complexes by a cluster of EAD-capable cells. A brief summary of the chapters is provided below: Chapter 2: In this chapter we investigate the effects of spatial gradients in the ion-channel conductances of various ionic currents on spiral-and scroll-wave dynamics. Ionic heterogeneities in cardiac tissue arise from spatial variations in the electrophysiological properties of cells in the tissue. Such variations, which are known to be arrhythmogenic, can be induced by diseases like ischemia. It is important, therefore, to understand the effects of such ionic heterogeneities on electrical-wave dynamics in cardiac tissue. To investigate such effects systematically, of changing the ion-channel properties by modifying the conductances of each ionic currents, on the action-potential duration (APD) of a myocyte cell. We then study how these changes in the APD affect the spiral-wave frequency ω in two-dimensional tissue. We also show that changing the ion-channel conductance not only changes ω but also the meandering pattern of the spiral wave. We then study how spatial gradients in the ion-channel conductances affect the spiral-wave stability. We find that the presence of this ionic gradient induces a spatial variation of the local ω, which leads to an anisotropic reduction of the spiral wavelength in the low-ω region and, thereby, leads to a breakup of the spiral wave. We find that the degree of the spiral-wave stability depends on the magnitude of the spatial variation in ω, induced by the gradient in the ion-channel conductances. We observe that ω varies most drastically with the ion-channel conductance of rapid delayed rectifier K+ current GKr, and, hence, a spiral wave is most unstable in the presence of a gradient in GKr (as compared to other ion-channel conductances). By contrast, we find that ω varies least prominently with the conductances of the transient outward K+ current Gto and the fast inward Na+ current (GNa); hence, gradients in these conduc-tances are least likely to lead to spiral-wave breaks. We also investigate scroll-wave instability in an anatomically-realistic human-ventricular heart model with an ionic gradient along the apico-basal direction. Finally, we show that gradients in the ion-channel densities can also lead to spontaneous initiation of spiral waves when we pace the medium at high frequency. Chapter 3: In this chapter we study the effects of gradients in the density of fibroblasts on wave-dynamics in cardiac tissue. The existence of fibroblast-myocyte coupling can modulate electrical-wave dynamics in cardiac tissue. In diseased hearts, the distribution of fibroblasts is heterogeneous, so there can be gradients in the fibroblast density (henceforth we call this GFD) especially from highly injured regions, like infarcted or ischemic zones, to less-wounded regions of the tissue. Fibrotic hearts are known to be prone to arrhythmias, so it is important to understand the effects of GFD in the formation and sustenance of arrhythmic re-entrant waves, like spiral or scroll waves. Therefore, we investigate the effects of GFD on the stability of spiral and scroll waves of electrical activation in a state-of-the-art mathematical model for cardiac tissue in which we also include fibroblasts. By introducing GFD in controlled ways, we show that spiral and scroll waves can be unstable in the presence of GFDs because of regions with varying spiral or scroll-wave frequency ω, induced by the GFD. We examine the effects of the resting membrane potential of the fibroblast and the number of fibroblasts attached to the myocytes on the stability of these waves. Finally, we show that the presence of GFDs can lead to the formation of spiral waves at high-frequency pacing. Chapter 4: In this chapter we study the arrhythmogenic effects of lo-calized fibrobblast clumps. Localized heterogeneities, caused by the regional proliferation of fibroblasts, occur in mammalian hearts because of diseases like myocardial infarction. Such fibroblast clumps can become sources of pathological reentrant activities, e.g., spiral or scroll waves of electrical activation in cardiac tissue. The occurrence of reentry in cardiac tissue with heterogeneities, such as fibroblast clumps, can depend on the frequency at which the medium is paced. Therefore, it is important to study the reentry-initiating potential of such fibroblast clumps at different frequencies of pacing. We investigate the arrhythmogenic effects of fibroblast clumps at high- and low-frequency pacing. We find that reentrant waves are induced in the medium more prominently at high-frequency pacing than with low-frequency pacing. We also study the other factors that affect the potential of fibroblast clumps to induce reentry in cardiac tissue. In particular, we show that the ability of a fibroblast clump to induce reentry depends on the size of the clump, the distribution and percentage of fibroblasts in the clump, and the excitability of the medium. We study the process of reentry in two-dimensional and a three-dimensional mathematical models for cardiac tissue. Chapter 5: In this chapter we investigate the role of early afterdepolarizations (EADs) on the formation of spiral and scroll waves. Early after depolarizations, which are abnormal oscillations of the membrane poten-tial at the plateau phase of an action potential, are implicated in the de-velopment of cardiac arrhythmias like Torsade de Pointes. We carry out extensive numerical simulations of the TP06 and ORd mathematical models for human ventricular cells with EADs. We investigate the different regimes in both these models, namely, the parameter regimes where they exhibit (1) a normal action potential (AP) with no EADs, (2) an AP with EADs, and (3) an AP with EADs that does not go back to the resting potential. We also study the dependence of EADs on the rate of at which we pace a cell, with the specific goal of elucidating EADs that are induced by slow or fast rate pacing. In our simulations in two- and three-dimensional domains, in the presence of EADs, we find the following wave types: (A) waves driven by the fast sodium current and the L-type calcium current (Na-Ca-mediated waves); (B) waves driven only by the L-type calcium current (Ca-mediated waves); (C) phase waves, which are pseudo-travelling waves. Furthermore, we compare the wave patterns of the various wave-types (Na-Ca-mediated, Ca-mediated, and phase waves) in both these models. We find that the two models produce qualitatively similar results in terms of exhibiting Na-Ca- mediated wave patterns that are more chaotic than those for the Ca-mediated and phase waves. However, there are quantitative differences in the wave patterns of each wave type. The Na-Ca-mediated waves in the ORd model show short-lived spirals but the TP06 model does not. The TP06 model supports more Ca-mediated spirals than those in the ORd model, and the TP06 model exhibits more phase-wave patterns than does the ORd model. Chapter 6: In this chapter we study the role of EAD-capable cells, and fibroblasts on the trigerring of abnormal electrical pulses called premature ventricular complexes (PVCs). Premature ventricular complexes, which are abnormal impulse propagations in cardiac tissue, can develop because of various reasons including early afterdepolarizations (EADs). We show how a cluster of EAD-generating cells (EAD clump) can lead to PVCs in a model of cardiac tissue, and also investigate the factors that assist such clumps in triggering PVCs. In particular, we study, through computer simulations, the effects of the following factors on the PVC-triggering ability of an EAD clump: (1) the repolarization reserve (RR) of the EAD cells; (2) the size of the EAD clump; (3) the coupling strength between the EAD cells in the clump; and (4) the presence of fibroblasts in the EAD clump. We find that, although a low value of RR is necessary to generate EADs and hence PVCs, a very low value of RR leads to low-amplitude EAD oscillations that decay with time and do not lead to PVCs. We demonstrate that a certain threshold size of the EAD clump, or a reduction in the coupling strength between the EAD cells, in the clump, is required to trigger PVCs. We illustrate how randomly distributed inexcitable obstacles, which we use to model collagen deposits, affect PVC-triggering by an EAD clump. We show that the gap-junctional coupling of fibroblasts with myocytes can either assist or impede the PVC-triggering ability of an EAD clump, depending on the resting membrane potential of the fibroblasts and the coupling strength between the myocyte and fibroblasts. We also find that the triggering of PVCs by an EAD clump depends sensitively on factors like the pacing cycle length and the distribution pattern of the fibroblasts.