  The task of this survey is to present various results on intersection
patterns of convex sets. One of main tools for studying intersection patterns
is a point of view via simplicial complexes. We recall the definitions of so
called $d$-representable, $d$-collapsible and $d$-Leray simplicial complexes
which are very useful for this study. We study the differences among these
notions and we also focus on computational complexity for recognizing them. A
list of Helly-type theorems is presented in the survey and it is also discussed
how (important) role play the above mentioned notions for the theorems. We also
consider intersection patterns of good covers which generalize collections of
convex sets (the sets may be `curvy'; however, their intersections cannot be
too complicated). We mainly focus on new results.
