A band saw is the ideal tool for cutting curves and slicing through thick stock. Smaller band saws should be able to slice though lumber 4" thick, with bigger saws being able to handle 6" cuts. Band saws are also capable of ripping, cross cutting, bevelling, and cutting aluminum, plastic and other materials.
For a saw with this much cutting capacity, you might expect a tool with an immense motor. But the band saw's efficient cutting action means it can get its muscle from a surprisingly compact source. Even on some of the biggest band saws, you'll rarely see a motor larger than 3/4 horsepower. Band saws come in both stationary models with cast-iron frames, and portable saws that can be moved easily and placed on a workbench or tabletop.