  In this paper we present some new results about unlike powers in arithmetic
progression. We prove among other things that for given $k\geq 4$ and $L\geq 3$
there are only finitely many arithmetic progressions of the form
$(x_0^{l_0},x_1^{l_1},...,x_{k-1}^{l_{k-1}})$ with $x_i\in{\Bbb Z},$
gcd$(x_0,x_1)=1$ and $2\leq l_i\leq L$ for $i=0,1,...,k-1.$ Furthermore, we
show that, for L=3, the progression $(1,1,...,1)$ is the only such progression
up to sign.
