a locally restricted composition of $n$ is a sequence of positive integers summing to $n$, where the allowed values of each part depend on the values of the $k$ preceding parts. for compositions of $n$ with $k=0$ (all compositions) or with $k=1$ and adjacent parts unequal (carlitz compositions) the number of compositions, behavior of the largest part and other information (e.g., number of parts) have been studied. i'll discuss \vskip .1in \noindent (a) similar results we have obtained for $k=1$ when the restriction is on the part difference and \vskip .1in \noindent (b) current research on the general case.\vskip .1in \noindent this is joint work with rod canfield and bill helton.