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In modern settings of data analysis, we may be running our algorithms ondatasets that are sensitive in nature. However, classical machine learning andstatistical algorithms were not designed with these risks in mind, and it hasbeen demonstrated that they may reveal personal information. These concernsdisincentivize individuals from providing their data, or even worse,encouraging intentionally providing fake data.  To assuage these concerns, we import the constraint of differential privacyto the statistical inference, considered by many to be the gold standard ofdata privacy. This thesis aims to quantify the cost of ensuring differentialprivacy, i.e., understanding how much additional data is required to performdata analysis with the constraint of differential privacy. Despite the maturityof the literature on differential privacy, there is still inadequateunderstanding in some of the most fundamental settings.  In particular, we make progress in the following problems:  $\bullet$ What is the sample complexity of DP hypothesis testing?  $\bullet$ Can we privately estimate distribution properties with a negligiblecost?  $\bullet$ What is the fundamental limit in private distribution estimation?  $\bullet$ How can we design algorithms to privately estimate random graphs?  $\bullet$ What is the trade-off between the sample complexity and theinteractivity in private hypothesis selection?

Statistical Inference in the Differential Privacy Model