Different approaches to the self-avoiding walk problem are reviewed. The problem first arose in the statistical physics of linear polymers in connection with the evaluation of the average size of a polymer. The probability distribution density $W_N(\mathbf{R})$ for the vector $\mathbf{R}$ connecting the end-points of an $N$-step self-avoiding walk is the main quantity in this problem. The equation for $W_N(\mathbf{R})$ seems to be invariant under the scaling transformation group. This means that the renormalization group method can be used to determine the asymptotic form of $W_N(\mathbf{R})$ as $N\to\infty$.