  We present an exposition of our ongoing project in a new area of applicable
mathematics: practical computation with finitely generated linear groups over
infinite fields. Methodology and algorithms available for practical computation
in this class of groups are surveyed. We illustrate the solution of hard
mathematical problems by computer experimentation. Possible avenues for further
progress are discussed. This article is aimed at a broad mathematical audience,
and more particularly at users of group-theoretical methods and computer
algebra systems.
