The author studies the complex spectra of second order elliptic systems \[ Lu\equiv -A_{ij}(x)u_{,ij}+B_ i(x),u_{,i}+C(x)u=\lambda M(x)u\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] \(\Omega \subset {\mathbb{R}}^ n\) a bounded \(C^{2+\theta}\)-domain, \(A_{ij}\), \(B_ i\), C, M are \(N\times N\)-matrices, \(A_{ij}\) satisfies \(Re a^{\alpha \beta}_{ij}(x) \xi^{\alpha}_ i {\bar \xi}^{\beta}_ j\geq a_ 0| \xi /^ 2\) (for some \(a_ 0>0)\) for all \(\xi \in {\mathbb{C}}^{nN}\), \(x\in {\bar \Omega}\); \(\lambda\in {\mathbb{C}}\) is the eigenvalue parameter. Under various conditions on L and M bounds are given for the real parts of the eigenvalues. The first part is based on integration by parts and the second one on the maximum principle for elliptic equations (with stronger structure conditions). This paper gives an extensive discussion of related results for elliptic equations.