We review the recent progress made in the application of the renormalization group method to the interacting quasi-one-dimensional electron gas. From the functional integral formulation of the partition function as expressed in terms of anticommuting Grassmann variables, the effects of single-electron interchain hopping on the mechanisms of propagation of correlations leading to long range ordering is analyzed in full details for the non-half-filled band case. Within a unified formalism, the scaling features of the purely one-dimensional correlations, the dimensionality crossover of both single and composite particles, the critical temperatures, and the responses functions are described.