  The main goal of this paper is to provide a brief survey of recent results
which connect together results from different areas of research. It is well
known that numerical integration of functions with mixed smoothness is closely
related to the discrepancy theory. We discuss this connection in detail and
provide a general view of this connection. It was established recently that the
new concept of {\it fixed volume discrepancy} is very useful in proving the
upper bounds for the dispersion. Also, it was understood recently that point
sets with small dispersion are very good for the universal discretization of
the uniform norm of trigonometric polynomials.

Numerical Analysis