Work that will be presented is in the field of "discrete-time quantum walks," which are generic quantum processes on a discrete space of states, with mixing of state components that is imposed at every step. The construct of a quantum walk has been established in quantum computing as an analog of classical random walks, targeted toward algorithms. However, it is also becoming a general investigative tool for physics unrelated to this original motivation. This talk will illustrate our belief that quantum walks are a flexible exploratory tool, along with being a powerful algorithmic approach. Most of presented work revolves around uses of a framework for construction of quantum walks that we introduced, that is unifying of other major approaches. I will first introduce the standard approach to discrete-time quantum walks. Then our formalism will be presented, and the body of the talk is about three different uses of this framework: To build and solve a walk on the binary tree; to examine walks in which (two-step) history is monitored; and, to pose and solve a simple-looking problem on matching graphs that in turn gives rise to a method for construction and analysis of complex networks. Broader ideas behind our work will be emphasized and kept track of throughout the discussion of specific problems. The talk is intended to make sense and be possible to follow without previous exposure to this field of research.