The results in this memoir are threefold. We consider a $c_0$-semigroup of operators acting on either a classical $L^p(\Omega)$ space, a Bochner $L^p(\Omega,Y)$ space or a noncommutative $L^p(M)$ space. First we are interested in functional calculus of the generator $A$, a question that has a long standing history since the fundamental works of Stein \cite{Ste70} and Cowling \cite{Cow} and is well-known to be of great importance among others in spectral theory, maximal regularity and control theory. Our spectral multipliers are then bounded holomorphic functions on a sector in the complex plane ($\HI(\Sigma_\omega)$ calculus) or defined on the positive half-line and coming with derivatives satisfying the H\"ormander Mihlin condition, i.e. are weighted bounded in a certain way ($\Hor^\alpha_2$ calculus). We show boundedness of the $\HI$ calculus for markovian and submarkovian semigroups on weighted $L^2(\Omega,wd\mu)$ space. Moreover, we obtain boundedness of $\Hor^\alpha_2$ calculus on $L^p(\Omega)$ for semigroups with Poisson estimates and for the Dunkl heat semigroup on the Bochner space $L^p(\R^d,h_\kappa^2 ,Y)$ where $Y$ is a UMD lattice, generalising work by Bonami-Clerc, Dai-Wang and Dai-Xu. Then we consider selfadjoint semigroups satisfying classical or generalised Gaussian estimates on spaces of homogeneous type and obtain $\Hor^\alpha_2$ calculus again on the UMD lattice valued Bochner space $L^p(\Omega,Y)$. In a second line, we are interested in maximal operators acting on $L^p(\Omega,Y)$, in its well-known spatial form of Hardy-Littlewood type or in spectral multiplier form $\sup_{t > 0} |m(tA)f|$. We obtain that the Fefferman-Stein inequalities hold with a dimension free bound, on the range $1 < p < \infty$, and with $Y = \ell^q,\:1 < q < \infty$ or more generally, $Y=$ a UMD lattice. We also obtain that the Hardy-Littlewood maximal operator is bounded on $L^p(\Omega,Y)$ with $\Omega$ a space of homogeneous type. For the spectral multiplier maximal operators, we obtain bounds on $L^p(\Omega,Y)$ for symbols $m$ decreasing at $\infty$, under the hypothesis of H\"ormander calculus of $A$ on $L^p(\Omega,Y)$. In a third part we investigate noncommutative Fourier and Schur multipliers and show among others dimension free bisectorial $\HI$ calculus of Hodge-Dirac operators stemming from markovian multiplier semigroups. This is built on previous work on Riesz transform estimates initiated by Junge, Mei and Parcet. These estimates also allow to prove that the Hodge-Dirac operators give rise to objects from noncommutative geometry: spectral triples and quantum (locally) compact metric spaces. Finally, in an independent section, we also investigate so-called decomposable operators acting on noncommutative $L^p$ spaces, which generalise pointwise boundedly dominated operators from classical $L^p$ theory. Then we consider again noncommutative Fourier and Schur multipliers and express their decomposability.