The spaces $\mathcal D_{L,\mu}^{(M_\alpha)}(\mathbf R)$ and $\mathcal B_\mu^{(M_\alpha)}$, $s\in[1,\infty)$, $\mu\in\mathbf R$, of weighted ultradifferentiable functions and the space $S^{(M_\alpha)}(\mathbf R)$ of rapidly decreasing ultradifferentiable functions on the real line $\mathbf R$ are defined. Their topological structure and their relations with $\mathcal D_{L^s}(\mathbf R)$, $\mathcal B(\mathbf R)$ and are investigated. The basic properties of the (ultra)differentiation, the multiplication and the convolution in the spaces are obtained. The space $O^{(M_\alpha)}_M(\mathbf R)$ of multipliers of the space $S^{(M_\alpha)}(\mathbf R)$ is determined.