We consider non-Markovian, self-interacting random walks on the one-dimensional integer lattice Z. The walk starts from the origin and at each step jumps to a neighbouring site, the probability of jumping along a bond being proportional to w(number of previous jumps along that lattice bond), where w : N ---> R_+ is a monotone weight function. Exponential and subexponential weight functions were considered in earlier papers. In the present paper we consider weight functions w with polynomial asymptotics. These weight functions define variants of the so-called `reinforced random walk'. We prove functional limit theorems for the local time processes of these random walks and local limit theorems for the position of the random walker at late times. A generalization of the Ray-Knight theory of local time arises.