1 Introduction
1.1 Simplicial complexes
1.2 d-representable complexes
1.3 What is in the survey?

2 d-collapsible and d-Leray complexes

3 Gaps among the notions

4 Algorithmic perspective
4.1 Representability
4.2 Collapsibility
4.3 Leray number
4.4 Greedy collapsibility

5 Good covers
5.1 Nerve theorems
5.2 Good covers versus collections of convex sets

6 Helly type theorems
6.1 The Helly theorem
6.2 The colorful Helly theorem
6.3 The fractional Helly theorem
6.4 The (p,q) theorem
6.5 The Amenta theorem