Vital cellular processes such as growth, gene expression, and homeostasis depend on the correct transmission of molecular signals within and between cells. The vast complexity of these molecular signaling networks has necessitated the use of mathematical methods to model, characterize, and predict cellular responses. The work presented in this dissertation shows how computational methods were used to elucidate two clinically-relevant cellular signaling responses: (i) phosphotyrosine signaling through the epidermal growth factor receptor (EGFR), a receptor tyrosine kinase that is commonly overexpressed or structurally altered in human cancers; and (ii) phosphoinositide and calcium signaling in human platelets---the key cellular mediators of hemostasis and pathological thrombus formation. The kinetic model of EGFR-mediated signaling in wild-type and mutant cells showed how mutant forms of the receptor use an irregular pattern of tyrosine phosphorylation that preferentially activates the survival oncoprotein, Akt. By quantifying the amount of signal flow through diverging pathways downstream of the receptor, our calculations provided a mechanistic explanation for the clinical observation that therapeutic tyrosine kinase inhibitors can control tumor growth in cells bearing certain EGFR mutations. In the second major study, a kinetic model of ADP-stimulated calcium release in human platelets was used to make precise, quantitative predictions about the molecular makeup and structural properties of the platelet. Specifically, we found that the resting structure of platelets places strong restrictions on several biophysical quantities, such as the resting concentration of free inositol 1,4,5-trisphosphate, the ratio of calcium ATPase pumps to release channels, and the size of the calcium storage compartment. Notably, the model also demonstrated that the irregular calcium spiking behavior observed in single ADP-stimulated platelets is due to the extremely small cellular volume. A novel method for constructing kinetic signaling networks, based on restricting the steady-state properties of the model, is also presented. Future applications and extensions of the systems approach to signal transduction modeling are discussed in the final chapter.