This article deals with various implicit and inverse function theorems for nondifferential maps, with constructive homeomorphism results for nonlinear and semilinear not necessarily differentiable maps, and with error estimates of approximable solutions. The class of considered maps can be characterized as operators having either some type of a multivalued derivative (in a Banach space) or a suitable variation (in a complete metric space). The article consists of 6 sections. Section 1 presents some survey of various generalizations of implicit function theorems and the general description of the basic results included in other sections. Section 2 deals with a row of new implicit function theorems based on using either topological or variational methods; the theorems of the section cover operator equations with pseudo-\(A\)-proper mappings, condensing (with respect to some measure of noncompactness) mappings, potential \(A\)-proper mappings and similar ones. Section 3 contains an approximation solvability result for nonlinear \(A\)-proper (with respect to a approximation scheme \(\Gamma\)) maps having multivalued derivative and the corresponding result on the error estimates for the approximate solutions; both theorems are important for the rest of the paper. Section 4 offers some criteria that a continuous \(A\)-proper mapping \(T\) turns out to be a homeomorphism and the equation \(Tx=f\) approximation-solvable (the corresponding error estimates are also presented). Section 5 deals with similar results on the unique approximation solvability for the nonresonant semilinear equations of type \(Ax- Nx=f\) in a Hilbert space, where \(A\) is a closed linear densily defined map and \(N\) is a nonlinear map such that \(A-N\) is \(A\)-proper. Section 6 deals with homeomorphism theorems and error estimates for approximation-stable \(A\)-proper maps. Some theorems presented in the article repeat earlier results of the author.