This paper is a collection of examples showing various effects on configurational properties of a random walk under the constraint that the ends of the random walk be fixed. Some formalism is developed for the solution of such problems, allowing the authors to express results in terms of the characteristic function of the unconstrained random walk. Several applications of this formalism are made in the paper. Typical of these is the result that fixing the ends of a random walk with an underlying stable-law density forces the steps of the resulting walk to have a number of finite moments. Other results show a range of differences to be expected between properties of constrained and unconstrained random walks.