Cardiac fibroblasts, when coupled functionally with myocytes, can modulate the electrophysiological properties of cardiac tissue. We present systematic numerical studies of such modulation of electrophysiological properties in mathematical models for (a) single myocyte-fibroblast (MF) units and (b) two-dimensional (2D) arrays of such units; our models build on earlier ones and allow for zero-, one-, and two-sided MF couplings. Our studies of MF units elucidate the dependence of the action-potential (AP) morphology on parameters such as Formula: see text, the fibroblast resting-membrane potential, the fibroblast conductance Formula: see text, and the MF gap-junctional coupling Formula: see text. Furthermore, we find that our MF composite can show autorhythmic and oscillatory behaviors in addition to an excitable response. Our 2D studies use (a) both homogeneous and inhomogeneous distributions of fibroblasts, (b) various ranges for parameters such as Formula: see text, and Formula: see text, and (c) intercellular couplings that can be zero-sided, one-sided, and two-sided connections of fibroblasts with myocytes. We show, in particular, that the plane-wave conduction velocity Formula: see text decreases as a function of Formula: see text, for zero-sided and one-sided couplings; however, for two-sided coupling, Formula: see text decreases initially and then increases as a function of Formula: see text, and, eventually, we observe that conduction failure occurs for low values of Formula: see text. In our homogeneous studies, we find that the rotation speed and stability of a spiral wave can be controlled either by controlling Formula: see text or Formula: see text. Our studies with fibroblast inhomogeneities show that a spiral wave can get anchored to a local fibroblast inhomogeneity. We also study the efficacy of a low-amplitude control scheme, which has been suggested for the control of spiral-wave turbulence in mathematical models for cardiac tissue, in our MF model both with and without heterogeneities.