We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 6 to 11 dimensions. They are between 20 to 10^4 times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing which allowed us to simulate clusters of millions of sites on computers with less than 500 MB memory; (ii) a histogram method which allowed us to obtain information for several p values from a single simulation; and (iii) a variance reduction technique which is especially efficient at high dimensions where it reduces error bars by a factor up to ~30. Based on these data we propose two new scaling laws, one for finite cluster size corrections and the other for asymptotic critical cluster growth speed as a function of dimension.