1 Introduction

2 Preliminaries  
2.1 Homology  
2.2 Geometric complexes  
2.3 Point processes  
2.4 Convergence of sequences of random variables  
2.5 Some notation  

3 Connectivity  
3.1 The sub critical regime  
3.2 The critical regime  
3.3 The super critical regime  

4 Homology and Betti Numbers  
4.1 The sub critical regime  
4.2 The critical regime  
4.3 The super critical regime  

5 Morse theory for the distance function  
5.1 Critical points of the distance function  
5.2 Morse Theory  
5.3 The Euler characteristic  

6 Extending to manifolds  
6.1 Closed Manifolds Embedded in $\mathbb{R}^{d}$  
6.2 Riemannian Manifolds and Homological Connectivity  

7 Stationary point processes

8 Extreme value analysis of random geometric complexes

9 Persistent homology  
9.1 Limit theorems for persistence diagrams  
9.2 Maximal cycles in persistent homology  

10 Open problems / future directions