The study of the density of norm attaining operators was initiated in the 1960's by J. Lindenstrauss who proved that there are bounded linear operators which cannot be approximated by norm attaining operators and also initiated the study of conditions assuring the density of norm attaining operators. Since then, a number of mathematicians have been working on this field as J. Uhl, J. Bourgain, W. Schachermayer, J. Partington, W. Gowers, R. Payá, M. Acosta... Our first aim in this talk will be to give an overview of the main results in this field. It has been very recently discovered that there are compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. The second aim of this talk will be to discuss those examples and also sufficient conditions to ensure that compact linear operators can be approximated by norm attaining operators.
 M. Martin, Norm-attaining compact operators, J. Funct. Anal. 267 (2014), 1585-1592. http://doi.org/10.1016/j.jfa.2014.05.019
 M. Martín, The version for compact operators of Lindenstrauss properties A and B, RACSAM 110 (2016), 269-284. http://doi.org/10.1007/s13398-015-0219-5