This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q In general these varieties are not compact The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G Af of finite adelic points of G The second action can be studied on the set of complex points of the Shimura variety In this book Sophie Morel identifies the Galois action at good places on the G Af isotypical components of the cohomology Morel uses the method developed by Langlands Ihara and Kottwitz which is to compare the Grothendieck Lefschetz fixed point formula and the Arthur Selberg trace formula The first problem that of applying the fixed point formula to the intersection cohomology is geometric in nature and is the object of the first chapter which builds on Morel s previous work She then turns to the group theoretical problem of comparing these results with the trace formula when G is a unitary group over Q Applications are then given In particular the Galois representation on a G Af isotypical component of the cohomology is identified at almost all places modulo a non explicit multiplicity Morel also gives some results on base change from unitary groups to general linear groups