Billinear forms with Kloosterman sums

We obtain new bounds on bilinear sums with Kloosterman sums and also 
with some similar sums. In particular, we improve some recent bounds on such bilinear sums due to V. Blomer, E. Fouvry, E. Kowalski, Ph. Michel and G. Mili{\'c}evi{\’c} (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 discuss further extensions of this method (still in progress, 
jointly with Tianping Zhang and Kui Liu) which improve some recent results of R. Nunes (2016). Finally, we outline some possible arithmetic applications of these bounds.