To obtain a characterization of Newton-like iterative methods with order of convergence at least three, we extend to Banach spaces the result given by Gander for scalar equations and mentioned in Chap. 3 . Many well-known third-order iterative methods are special cases of Gander’s result: the Chebyshev method, the Super-Halley method, the Halley method, Chebyshev-like methods, the Ostrowski method, the Exponential method, the Logarithmic method, and the Euler method. From the last methods, we see that we can generalize them to obtain a family of iterative methods that includes them. In addition, we see that all methods with R-order of convergence at least three in Banach spaces also admit a general expression. Our aim is to obtain a theory, the most general possible, relative to these iterative methods in Banach spaces which analyze their semilocal convergence under Kantorovich-type conditions for iterative methods with R-order of convergence at least three. For this, we use a technique based on recurrence relations. We then relax the last required conditions and, using the method of majorizing sequences, prove the semilocal convergence of the family of iterative methods defined previously. Finally, we analyze the semilocal convergence of a particular family of iterative methods, which is included in the previous family, whose R-order of convergence is at least four when applied to solve quadratic equations. We also use the theoretical significance of the methods to draw conclusions about the existence and uniqueness of solution of the equation to solve and illustrate all the theoretical results with examples.