This paper studies the critical and near-critical regimes of the planar random-cluster model on \mathbb Z^2 with cluster-weight q\in[1,4] using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents \beta, \gamma, \delta, \eta, \nu, \zeta as well as \alpha (when \alpha\ge0). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalization of Kesten's classical scaling relation for Bernoulli percolation involving the "mixing rate" critical exponent \iota replacing the four-arm event exponent \xi_4.
