Pressure-sensitive adhesives containing at least one functionalized polymer or block polymer having a weight average of the molar mass distribution of between 5,000 g/mol and 200,000 g/mol and a difference between Mp and Mmin which is lower than 25,000 g/mol, where Mp represents the peak molar mass of a monomodal polymer or the peak molar mass of the smallest mode in a bimodal or multimodal polymer and Mmin represents the lowest molar mass in the molar mass distribution, defined at each point where the cumulative frequency of the molar mass distribution has a value of 0.05, the functionalized polymer or block polymer carries at least one type of functionalization that can be used for a subsequent cross-linking and the at least one type of functionalization of the functionalized polymer results from the use of at least one type of functionalized monomer during the production of the functionalized polymer.