We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK inequality. For our main results, we consider three versions of the connection function $\varphi$: a finite-variance version (including the Boolean model), a spread-out version, and a long-range version. For sufficiently large dimension (resp., spread-out parameter and $d>6$), we then prove the convergence of the lace expansion, derive the triangle condition, and establish an infra-red bound. From this, mean-field behavior of the model can be deduced. As an example, we show that the critical exponent $γ$ takes its mean-field value $γ=1$ and that the percolation function is continuous.