1. **State the problem:** We need to find the SI units of the constant $G$ in Newton's law of universal gravitation given by $$F=\frac{GMm}{r^2}.$$ Here, $F$ is force with units $\mathrm{kg\cdot m/s^2}$, $M$ and $m$ are masses with units $\mathrm{kg}$, and $r$ is distance with units $\mathrm{m}$. We want the units of $G$. 2. **Write the equation for the units:** Substituting units, $$[F] = \frac{[G][M][m]}{[r]^2}.$$ 3. **Express units:** $$\mathrm{kg\cdot m/s^2} = \frac{[G] \times \mathrm{kg} \times \mathrm{kg}}{\mathrm{m}^2}.$$ 4. **Solve for $[G]$:** Multiply both sides by $\mathrm{m}^2$, $$[G] = \frac{\mathrm{kg\cdot m/s^2} \times \mathrm{m}^2}{\mathrm{kg}^2} = \frac{\mathrm{kg\cdot m^3/s^2}}{\mathrm{kg}^2}.$$ 5. **Simplify units:** $$[G] = \mathrm{m^3/(kg\cdot s^2)}.$$ **Final answer:** The SI units of $G$ are $$\boxed{\mathrm{m^3/(kg\cdot s^2)}}.$$