This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are

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... Key words. Systems of nonlinear equations, systems of polynomial equations, simplicial algorithms, pivoting algorithms, homotopy algorithms, Kuhn's method for finding roots of a polynomial, Nielsen fixed point theory.