  Fitting percolation into the conformal field theory framework requires
showing that connection probabilities have a conformally invariant scaling
limit. For critical site percolation on the triangular lattice, we prove that
the probability that $n$ vertices belong to the same open cluster has a
well-defined scaling limit for every $n \geq 2$. Moreover, the limiting
functions $P_n(x_1,\ldots,x_n)$ transform covariantly under M\"obius
transformations of the plane as well as under local conformal maps, i.e., they
behave like correlation functions of primary operators in conformal field
theory. In particular, they are invariant under translations, rotations and
inversions, and $P_n(sx_1,\ldots,sx_n)=s^{-5n/48}P_n(x_1,\ldots,x_n)$ for any
$s>0$. This implies that $P_{2}(x_1,x_2)=C_2 \Vert x_1-x_2 \Vert^{-5/24}$ and
$P_3(x_1,x_2,x_3) = C_3 \Vert x_1-x_2 \Vert^{-5/48} \Vert x_1-x_3 \Vert^{-5/48}
\Vert x_2-x_3 \Vert^{-5/48}$, for some constants $C_2$ and $C_3$.
  We also define a site-diluted spin model whose $n$-point correlation
functions $\mathrm{C}_{n}$ can be expressed in terms of percolation connection
probabilities and, as a consequence, have a well-defined scaling limit with the
same properties as the functions $P_n$. In particular,
$\mathrm{C}_{2}(x_1,x_2)=P_{2}(x_1,x_2)$. We prove that the magnetization field
associated with this spin model has a well-defined scaling limit in an
appropriate space of distributions. The limiting field transforms covariantly
under M\"obius transformations with exponent (scaling dimension) $5/48$. A
heuristic analysis of the four-point function of the magnetization field
suggests the presence of an additional conformal field of scaling dimension
$5/4$, which counts the number of percolation four-arm events and can be
identified with the so-called "four-leg operator'' of conformal field theory.
