The family of twisted $L$-functions of a cusp form

We will present some of the results recently obtained concerning the family of $L$-functions of a fixed cusp form twisted by Dirichlet characters of large prime modulus, and we will attempt to indicate some of the key ideas in the proofs. These include new results on bilinear forms with Kloosterman sums, which in turn involve delicate arguments with the cohomological interpretation of exponential sums over finite fields. (Joint works with V. Blomer, D. Milićević, É. Fouvry, Ph. Michel and W. Sawin).