This thesis describes new methods in input/output systems theory. The first part of this thesis develops a novel graphical method for the stability analysis of feedback systems. The Scaled Relative Graph (SRG), a recent concept from monotone operator theory, is shown to generalize the classical Nyquist diagram of an LTI transfer function, and may be plotted for arbitrary nonlinear operators. Applying the SRG to the analysis of feedback interconnections leads to a graphical incremental stability theorem, which unifies and generalizes the Nyquist criterion, circle criterion, incremental passivity theorem, incremental small gain theorem and incremental secant condition. A novel special case of this theorem concerns systems which only violate the assumptions of the incremental passivity theorem when their incremental gains are small. This captures systems whose incremental passivity is destroyed by common effects such as delay and saturation. The second part of this thesis develops algorithmic methods for solving nonlinear circuits composed of monotone elements. Monotonicity is a generalization of the linear property of passivity, and is a fundamental property in the theory of large-scale convex optimization. Modern splitting methods, which invert sums of operators, are shown to correspond to the series or parallel interconnection of circuit elements which are monotone, and may be used to solve the steady-state behavior of such circuits. Consideration of circuits with arbitrary series/parallel interconnections leads to a new splitting algorithm, the nested forward-backward algorithm, which inverts operators composed of sums and inverses. Finally, these methods are extended to circuits composed of the difference of monotone elements. The steady-state behavior of such circuits can be solved via an adaptation of Difference of Convex (DC) Programming. Such circuits include the classical van der Pol oscillator and the FitzHugh-Nagumo model of an excitable neuron, which both consist of an LTI transfer function in parallel with monotone and anti-monotone nonlinear resistors. A new algorithm, the difference-of-monotone Douglas-Rachford algorithm, is proposed, which matches the mixed-feedback structure of these circuits.