In this talk, we will first briefly review results related to arithmetic functions in short intervals, then we will focus on methods of the celebrated Matomaki-Radziwill theorem which shows that $$ \sum_{x< n \leq x+h } \mu(n) = o(h) $$ holds for almost all $x \in [X,2X]$, where $h \to \infty$ as $X \to \infty$. If time permits, we will introduce the recent breakthrough on primes in short intervals given by Maynard and Guth, where they proved that $$ \pi(x+h)-\pi(x) \sim \frac{h}{\log x} $$ for sufficiently large $x$ and $x^{17/30+ \epsilon}< h \leq x$, improved on Huxley’s results from more than 50 years ago.