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Roughing gouge - a wide fluted gouge used to initially round a wooden spindle, and to roughly shape it. Generally not intended for cutting end grain due to the large cut it takes and the relatively weak tang connecting the blade to the handle. Unsafe for making bowls or any faceplate work.
 
spindle gouge or detail gouge - a shallow fluted gouge used to create details on spindles, including beads (raised portions of the turning typically semi-circular in cross section) and coves (relieved portions of the turning).

bowl gouge - a deep fluted gouge used to turn the outside and inside of bowls and vessels. Often has a thicker shaft and longer handle than a spindle gouge because it has to cut farther away from the handrest.
 
skew chisel - a wide, steeply pointed chisel with the edge running at an angle to the length of the tool. Used to smooth flat spindles, cut beads, and add details. Skew chisels are only used on spindle work (never on faceplate work) and are honed after sharpening to create a razor edge.
 
parting tool - a pointed tool used to separate (part off) work from the lathe, and to create a straight edge separating large and small diameter sections - wide parting tools also called bedans are used to create evenly sized spindle sections.
 
hollowing tool - many different types of tools used to cut out the deep sections of steep bowls, vases and hollow vessels. Often with very long handles, to maintain enough leverage when working in a deep vessel, far away from the handrest.
 
scraper - a tool that scrapes the wood fibers instead of cutting - these are used to smooth off wooden items cut with other tools, and to shape items that are not possible or difficult to shape with gouges. A sharp scraper has a burr at the edge which cuts the wood, only a dull scraper actually scrapes.

bowl saver - a tool used to core out the inside part of a bowl, allowing the waste piece to be used to create a smaller bowl, and to limit the amount of wood chips created when hollowing out a bowl.
auger - a drill bit used to drill a hole partway or all the way through a wooden item. For cutting the hole for a lamp cord, or as the first step when hollowing out a bowl or vessel.
 
chatter tool - a flexible scraper used to add decorative chatter marks to turned items.
 
wire - a simple wire, sometimes with handles attached at either side, for the purpose of burning lines into the piece with friction.
There are also several tool types for special purposes, as well as tools that are a combination design of the above tools, i.e. skew/chisel combinations, thread cutting tools, ring cutting tools, medium fluted gouges, etc

There new tools coming on the market all the time. To know which ones are useable and of good quality, seek the advice of a fellow woodturner who has been active in the field for awhile. Making your own woodturning tools has become very popular among turners recently. A lot of times, there is just no tool for a specific application or turning style so a custom made tool may be just the ticket. Learning how to make turning tools is not that difficult and information can be found on all the wood turning forums.

Turning tools are generally made from three different types of steel, Carbon steel, High speed steel (HSS), and more recently Powdered Metal. Comparing the three types, high speed steel tools maintain their edge longer, requiring less frequent sharpening than carbon steel, but not as long as powdered metal tools. The harder the type of high speed steel used, the longer the edge will maintain sharpness. Powdered steel is even harder than HSS, but does not take an edge that is as sharp as HSS. Unlike other edged woodworking tools, woodturning tools require more frequent sharpening, because the wood passes at a great speed. To maintain a clean cut, the sharpness of the tools edge must be maintained. The sharpening process requires either skill of the craftsman, or one of the many available sharpening jigs, which facilitate maintaining a specific bevel on the tool.