In this paper, we provide some methods for generating slant helices lying on a parametric or an implicit surface in Euclidean 3-space. These methods are based on the initial value problems of first order ordinary differential equations. Although analytical solutions of the derived systems are not possible even for simple surfaces, we provide some illustrative examples in which the slant helices have been obtained numerically. The produced slant helices possess the given fixed direction, constant angle, and the curvature function. (C) 2019 Elsevier Inc. All rights reserved.