  In this paper we show a general method to compactify certain open varieties
by adding normal crossing divisors. This is done by proving that {\it blowing
up along an arrangement of subvarieties} can be carried out. Important examples
such as Ulyanov's configuration spaces, spaces of holomorphic maps, etc., are
covered. Intersection ring and (non-recursive) Hodge polynomails are computed.
Further general structures arising from the blowup process are described and
studied.
