The field of topological data analysis has received attention in the recent years and a key object of study are random simplicial complexes. After a brief introduction to the field, we shall look at the more specific case of simplicial complexes built over random points in space. We shall in particular try to see how dependence structure of the random points affect the topology of the random simplicial complexes. We shall also try to illustrate the proof techniques rely on the study of the corresponding the more well-known random geometric graphs. Only a basic knowledge of probability and topology will be assumed (this is a joint work with Prof. Robert Adler).