Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension $d\geq2$. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for $d=2$) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter $\lambda$. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of $\lambda$ that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions $d\geq2$. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone. Furthermore, for $d=2$ we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.