The history of using nonconforming finite elements for fluid and solid mechanics goes back to 1960’s. We will briefly review recent developments on nonconforming finite element methods. Mathematical analysis and numerical results will be demonstrated to compare conforming and nonconforming finite element methods. We also introduce a cheapest stable nonconforming finite element pair for Stokes equations. The finite element pair has lowest degrees of freedom with optimal order of convergence. We then apply the elements and other known conforming and nonconforming finite element pairs to driven cavity flows. Three types of indicators for the accuracy checking will be addressed. Finally numerical comparison of these finite element pairs are presented based on the indicators. Other applications will be presented solving multiscale problems related with Darcy’s flow.