Abstract: Flow past circular cylinders and square cylinders at high Reynolds numbers are simulated by improved delayed detached-eddy simulation (IDDES), including a circular cylinder, a square cylinder, two tandem circular cylinders and two tandem square cylinders. The mean drag coefficient, the RMS values of lift coefficient and the Strouhal number are computed for various Reynolds numbers, which show a good agreement with previous experimental and numerical simulation data. It is found that the effect of Reynolds number on the global quantities for square cylinders is not much in this range of Reynolds numbers, which is different for circular cylinders. There is no drag crisis phenomenon for flow past a square cylinder at $2.0\times 10^3<\mathit{Re}<1.0\times 10^7$. The Strouhal number is Reynolds-independent for $\mathit{Re}>2.0\times 10^3$, and the Reynolds-independent is also observed for the mean drag coefficient and the RMS lift coefficient. Simulation for two tandem circular cylinders is performed at Reynolds numbers of $2.2\times 10^4$ and $3.0\times 10^6$ for five different spacing $L$ to diameter $D$ ratios: $L/D=2.0$, 2.5, 3.0, 3.5 and 4.0. At the critical spacing ($L_c/D)$ there is found a distinct step-like jump of mean drag coefficient and RMS lift coefficient of the subcritical Reynolds number of $2.2\times 10^4$, and the mean drag coefficient of the downstream circular cylinder is negative for $L/D<L_c/D$. However, the mean drag coefficient and the RMS lift coefficient are seen to be slightly affected by spacing for $\mathit{Re}=3.0\times 10^6$, and the mean drag coefficient of the downstream circular cylinder is always positive. Flow past two tandem square cylinders is considered at Reynolds numbers of $1.6\times 10^4$ and $\mathit{Re}=1.0\times 10^6$. The abrupt change in mean drag coefficient and RMS lift coefficient at the critical spacing is clearly seen on both upstream and downstream square cylinders for both Reynolds numbers. When $L/D<L_c/D$, the mean drag coefficient of the downstream cylinder is negative for both Reynolds numbers.