The power of uniform design (UD) has received great attention in the area of computer experiments over the last two decades. However, when conducting a typical computer experiment, one finds many non-rectangular types of input domains on which traditional UD methods cannot be adequately applied. In this study, we propose a new UD method that is suitable for any type of design area. For practical implementation, we develop an efficient algorithm to construct a so-called nearly uniform design (NUD) and show that it approximates very well the UD solution for small sizes of experiment. By utilizing the proposed UD method, we also develop a methodology for estimating the target region of computer experiments. The methodology is sequential and aims to (i) provide adaptive models that predict well the output measures related to the experimental target; and (ii) minimize the number of experimental trials. Finally, we illustrate the developed methodology on various examples and show that, given the same experimental budget, it outperforms other approaches in estimating the prespecified target region of computer experiments. (C) 2009 Elsevier B.V. All rights reserved.