We present the first local list-decoding algorithm for the rth order 
Reed-Muller code RM(r,m) over F_2 for r>1.  Given an oracle for a received 
word R: F_2^m to F_2,  our randomized local list-decoding algorithm produces 
a list containing all degree r polynomials within relative distance  
2^{-r} - eps from R for any eps > 0 in time poly(m^r,eps^{-r}). The list 
size could be exponential in m at radius 2^{-r}, so our bound is optimal in 
the local setting. Since RM(r,m) has relative distance 2^{-r}, our algorithm
beats the Johnson bound for r>1. 

In the setting where we are allowed running-time polynomial in
the block-length, we show that list-decoding is possible up to even
larger radii, beyond the minimum distance. We give a deterministic
list-decoder that works at error rate below J(2^{1-r}), where
J(d) denotes the Johnson radius for minimum distance d. This shows that 
RM(2,m) codes are list-decodable up to radius s for any constant s < 1/2
in time polynomial in the block-length.  

Over small fields F_q, we present list-decoding algorithms in both the 
global and local settings that work up to the list-decoding radius. 
We conjecture that the list-decoding radius approaches the minimum
distance (like over F_2), and prove this when the degree is divisible 
by q-1.