An adaptive finite element method is given for approximating solutions to the Laplace-Beltrami 
equation on surfaces in $\mathbb{R}^3$ which may be implicitly represented as level sets of smooth 
functions.  Residual-type a posteriori erro bounds which show that the error may be split into a 
"residual part" and a "geometric part" are established.  In addition, implementation issues are 
discussed and several computational examples are given.