The Rudin-Shapiro coefficients $\{a(n)\}$ are an infinite sequence of $\pm 1$'s, defined recursively by $a(0)=1$, $a(2n)=a(n)$, and $a(2n + 1)=(-1)^{n}a(n)$, $n \geq 0$ Various formulas are developed for the $n$th partial sum $s(n)$ and the $n$th alternating partial sum $t(n)$ of this sequence. These formulas are then used to show that $\sqrt{3/5} < s(n)/\surd n < \surd 6$ and $0 \leq; t(n)/\surd n < \surd 3$, $n \geq 1$ where the inequalities are sharp and the ratios are dense in the two intervals. For a given $n \geq 1$, the equation $s(k)= n$ is shown to have exactly $n$ solutions $k$.