We outline some new bounds on bilinear sums with Kloosterman sums and also with some similar sums. In particular, these bounds improve some recent results of V. Blomer, E. Fouvry, E. Kowalski, Ph. Michel and G. Milicevic (2014-2016). As a result we improve the error term in the asymptotic formula for mixed moments of $L$-series associated with Hecke eigenforms.
We also use the same method to improve, in some ranges, a bound of S. Bettin and V. Chandee on trilinear sums with short incomplete Kloosterman sums.
Finally, discuss further extensions of this method (jointly with Tianping Zhang and Kui Liu), which improve some recent results of R. Nunes (2016) and outline some possible arithmetic applications of these bounds.