This talk will address the challenge of complexity in numerical simulations of complex physical problems.When the latter exhibit a multiscale and or a multiphysics nature, appropriate mathematical models and accurate numerical methods are required to catch the essential features of the manifold components of the physical solution. Often the associated numerical problem is so large that devising computational reduction techniques, and developing efficient parallel algorithms by exploiting a dimensional reduction paradigm, becomes necessary.  This presentation will elaborate on some of these issues and illustrate a few representative applications to diverse fields such as fluid and solid mechanics, including engineering and life sciences.  More in particular, some general purpose numerical strategies, based on model order reduction, geometrical and algebraic reduction, and heterogeneous domain decomposition, will be reviewed and used with the aim of drastically reducing the computational complexity.