Floating Point 

The main problem in floating point numbers being the only numeric type in Lua is that most programmers do not understand floating point. Floating point numbers are A LOT more complicated than integers (fixed point numbers). Most programmers' mental model of floating point is that the result of an operation sometimes has a very slight error. As a consequence, people get jittery. But this mental model is wrong; modeling floating point arithmetic as "correct answer + noise" is wrong, particularly when dealing with the most common type of floating point, IEEE754.
The solution to this whole problem is education. The essential integerbackwardscompatible behavior of double floating point should be emphasized more in the documentation, the website, the FAQ, the wiki, etc.
Before going further, it should be noted that although floating point numbers are often used as the numeric type in Lua, Lua does not depend on floating point but can be compiled with any numeric type, such as single precision floating point, integer, or even some very unfamiliar numerical representations[*5]. Of course, the semantics of division, for example, changes significantly if you compile Lua with an integer numeric type, but on an embedded processor that would probably not come as a surprise.[*1] ARM, Coldfire, and various flavors of embeded MIPS have no FPU, yet these are _very_ widely used.[*2]
Here are some points about (IEEE 754) 64bit doubleprecision floating point numbers ("doubles") v.s. 32bit integers:
Summary: 64bit precision floating point arithmetic is better than 32bit integer arithmetic, from the point of view of accuracy, range, and 'backwardscompatibility'. See the caveats section though.
IEEE 754 defines the basic mathematical operations, +  * (multiplication) / (division), to produce results that are as close as possible to the exact mathematical answer. In particular, if the answer is exactly representable as a doubleprecision floating point then that will be the answer; otherwise, there are two doubleprecision floating point numbers that are nearest the exact answer (one just above and one just below), and one of those two is selected according to the rounding mode.
For operations involving integers, most of the operations are exact. If I compute 2+5, then because the exact mathematical answer is 7, that is the also IEEE 754 answer.
Admittedly Intel's Pentium design comes a poor third (because it has too few FP registers). But Intel is catching up.
The only remaining performance concern is floating point to integer conversion. Like it or not, memory load instructions operate with an address and byte offset (i.e. two integer values). Therefore, any performance savings of using floating point instead of integers is for naught if the CPU's floattoint conversion performance is poor. Some claims state that float to int conversion can have a devastating effect on performance [1]. (For gcc3.4 and a circa 2006 AMD processor, the problem is gone, but test it for yourself. It is easy to compile the benchmark from that link.)
For users of these serious modern desktop CPUs the only major remaining potential issues are memory bandwidth and memory usage. Presumably, because of the cell (union structure) size in Lua for these objects, these considerations are actually irrelevant.
Hard data about this would be good here, such as the sizeof()
the appropriate union with single and double precision floating point configed under various (or one) architecture(s).
 In the world of 8byte structure alignment, they don't really take any more space, though if the object structure ever gets over 64 bits due to the size of a FP number, the size issue is "durn tootin" relevant. FP numbers also take more room in CPU caches  it's not just registers that suffer.  ChuckAdams
Summary: Circa 2001 your average entry level $10 CPU chip is capable of computing with 64b double as fast and as accurately as integer arithmetic (or more so). Users of these CPUs should not pay any costs by virtue of using double floating point only in lua, and may reap benefits. Users of smaller chips may well be obliged to use integer arithmetic only in lua.
 as also noted below, most ARM chips, one of the most widely used embedded CPUs, have no hardware floating point support. This means every conversion between int and float, and every arithmetic operation, is a function call (usually inlined). Doubles are even worse. When you consider the checks for NaN and denormalized numbers, the overhead vs. integers is quite significant (and yes, I have profiled). BenStJohn?
While Lua's number type can be built as integer, profiling is suggested before use since (even on integeronly processors) the speed up might be not as great as imagined and the loss of floating point can make some operations more complex. Also note Lua 5.3 has explicit integer support built in.
eLua http://www.eluaproject.net/ is a fork of Lua for some microcontrollers which has an optional integer build. However, it should be noted that standard Lua also builds on cores like the ARM7TDMI quite easily  since the source is standard C. The integer changes can be applied if necessary.
printf
implementations may not be able to handle printing floating point numbers accurately. Believe it or not, some may incorrectly print integers (that are floating point numbers). This can manifest itself as incorrectly printing some numbers in Lua. QNX has this problem [2]. The only sensible fix is to complain to your vendor and/or replace printf
with one that works.
These are not in any particularly relevant order:
 MartinHollis (original author)