This thesis is devoted to the study of contraction semigroups generated by linear partial differential operators. It is shown that linear partial differential operators of order higher than two cannot generate contraction semigroups on (L^P)^N for p ∈ [1, oo) unless p = 2. If p > 1 and the L^P-dissipativity criterion is restricted to the cone of nonnegative functions for differential operators with real-valued coefficients, it is proven that the criterion still fails for operators of order higher than two, except for some fourth order operators if ≤ p ≤ 3. A class of such fourth order operators is also presented. In connection with this, an auxiliary approximation result is given, which is of interest in itself: any nonnegative function, belonging to a class of Sobolev functions within complete zero Cauchy data, can be approximated by smooth nonnegative functions. For operators associated with systems uniformly parabolic in the sense of Petrovskii, necessary and sufficient conditions in algebraic terms are presented in order for the generation of contraction semigroups on (L^P)^N for all p ∈ [1, oo] simultaneously. A class of weakly coupled systems is studied and a close relationship between the generation of(L^P)^N-contractive semigroups of the corresponding operators and (L²)^N-contractivity of the semigroups generated by some associated operators is obtained. More precisely it is seen that (L²)^N-dissipativity of an associated operator implies (L^P)^N-dissipativity of the original operator, whereas the converse holds for some subclass of operators, which includes the scalar operators.