In this thesis spectral inequalities and trace formulae for discrete and continuous differential operators are discussed. We first investigate spectral inequalities for Jacobi operators with matrix-valued potentials and present a new, direct proof of a sharp inequality corresponding to a Lieb-Thirring inequality for the power 3/2 using the commutation method. For the special case of a discrete Schrödinger operator we also prove new inequalities for higher powers of the eigenvalues and the potential and compare our results to previously established bounds. We then approximate a Schrödinger operator on L²(R) by Jacobi operators on ℓ²(Z) and use the established inequalities to provide new proofs of sharp Lieb-Thirring inequalities for the powers γ = 1/2 and γ = 3/2. By means of interpolation we derive spectral inequalities for Jacobi operators that yield (non-sharp) Lieb-Thirring constants on the real line for powers 1/2 < γ < 3/2. We then consider Schrödinger operators on a finite interval [0,b] with matrix-valued potentials and establish trace formulae of the Buslaev-Faddeev-Zakharov type. The results link sums of powers of the negative eigenvalues to terms dependent on the potential and scattering functions. Finally, we discuss the Berezin inequality, which is well-known on sets of finite measure and find an analogous inequality for the magnetic operator with constant magnetic field on a set whose complement has finite measure. We obtain a similar bound for the Heisenberg sub-Laplacian.