  The challenge of mastering computational tasks of enormous size tends to
frequently override questioning the quality of the numerical outcome in terms
of accuracy. By this we do not mean the accuracy within the discrete setting,
which itself may also be far from evident for ill-conditioned problems or when
iterative solvers are involved. By accuracy-controlled computation we mean the
deviation of the numerical approximation from the exact solution of an
underlying continuous problem in a relevant metric, which has been the
initiating interest in the first place. Can the accuracy of a numerical result
be rigorously certified - a question that is particularly important in the
context of uncertainty quantification, when many possible sources of
uncertainties interact. This is the guiding question throughout this article,
which reviews recent developments of low-rank approximation methods for
problems in high spatial dimensions. In particular, we highlight the role of
adaptivity when dealing with such strongly nonlinear methods that integrate in
a natural way issues of discrete and continuous accuracy.
