I would like to remove the numbering on the axes of the following RegionPlot. I would like to keep the tick marks but drop the numbering, I haven't figured out how to do this from the documentation.

I would like to remove the numbering on the axes of the following RegionPlot. I would like to keep the tick marks but drop the numbering, I haven't figured out how to do this from the documentation.

An even simpler way that does not require you to figure out the tick positions, is to set the tick font opacity to 0 and the tick font size to 0 to avoid the excess margin where the ticks would have been. Here's an example:
RegionPlot[Sin[x y] > 0, {x, -1, 1}, {y, -1, 1}, FrameTicksStyle -> Directive[FontOpacity -> 0, FontSize -> 0]] 
Alternately, you could also use FontColor -> White, but note that it won't work with all backgrounds.
FontSize -> 0 in my test code, but threw it out in the end. I've included it now and made FontOpacity the primary suggestion (it certainly is superior to FontColor) $\endgroup$ My preferred solution is playing with FontColor or FontOpacity as in R.M.`s answer, or define your own ticks as in David's answer.
Another alternative is to change the labels to blank in FrameTicks. Since FrameTicks->Automatic saves a lot manual effort (and it uses the built-in FindDivisions for selecting ticks), sometimes it may be more convenient to transform the automatic frameticks data:
rp = RegionPlot[x^2 + y^3 < 2 && x + y < 1, {x, -2, 2}, {y, -2, 2}, FrameTicks -> Automatic] 
First extract the frameticks information and change the labels to blank:
newticks = Last@First[AbsoluteOptions[rp, FrameTicks]]; newticks[[All, All, 2]] = ""; Then replot or useShow with the new frameticks:
Show[rp, FrameTicks -> newticks] to get:

newticks[[All, All, 2]] = Spacer[0]; $\endgroup$ " ". But using Spacer[k] gives more precise control. $\endgroup$ ImageSize and set AspectRatio to Full in that way then I can specify my ImagePadding to correctly position my plot. I like your answer since it removes the unwanted data instead of just hiding it by making it transparent. The problem I have now is that I cannot extract the ticks marks because it wants the AspectRatio to be something other than Full. Any ideas how to deal with this? $\endgroup$ AbsoluteOptions[rp, FrameTicks] produces error: "Ticks::ticks: "{Automatic,Automatic} is not a valid tick specification." Odd error. $\endgroup$ I don't believe this has been mentioned yet:
RegionPlot[1 < Abs[x + I y] < 2, {x, -2, 2}, {y, -2, 2}, ImagePadding -> 1] 
ImagePadding
is an option for graphics functions that specifies what absolute extra padding should be left for extended objects such as thick lines and annotations such as tick and axis labels.
ImagePadding specifies the space around the graph outside of the Frame. Normally this is calculated to include the ticks and labels. I tried 0 but that cuts into the frame slightly, so 1 crops everything but the frame. $\endgroup$ Frame (precisely because you're using ImagePadding). For example, Graphics[{Thickness[.3], Pink, Circle[]}, ImagePadding -> 1, Frame -> True] (remove ImagePadding to see how it ought to look) $\endgroup$ FrameTicks
Consider FrameTicks in the example below:
RegionPlot[Sin[x] Sin[y] > 1/4, {x, -1, 1}, {y, -1, 1}, FrameTicks -> {Table[{k, "", {If[Mod[k, .5] == 0, .02, .01], 0}}, {k, -1, 1, .1}], Table[{k, "", {If[Mod[k, .5] == 0, .02, .01], 0}}, {k, -1, 1, .2}]}] 
In Table[{k, "", {If[Mod[k, .5] == 0, .02, .01], 0}}, {k, -1, 1, .1}] ,
k is the location of the tick
"" is the label
If... determines whether to use a major (.02) or minor (.01) tick on the inside of the respective axis.
0 indicates that the tick should not appear on the outside of the axis.
Note that the first Table sets the horizontal edges of the frame (with ticks every .1 units); the second Table sets the vertical edges of the frame (with ticks every .2 units).
You can use the internal, undocumented function Charting`ScaledFrameTicks to do this (Charting`ScaledFrameTicks produces unlabeled ticks, while Charting`ScaledTicks produces labeled ticks):
RegionPlot[ Sin[x y]>0, {x,-1,1}, {y,-1,1}, FrameTicks->{ {Charting`ScaledFrameTicks[{Identity,Identity}], Automatic}, {Charting`ScaledFrameTicks[{Identity,Identity}], Automatic} } ]