We prove sharp maximal inequalities for L q -valued stochastic integrals with respect to any Hilbert space-valued local martingale. Our proof relies on new Burkholder-Rosenthal type inequalities for martingales taking values in an L q -space.Contents to be sharp or even to depend optimally on p and q. For applications to stochastic evolution equations, the precise constants in fact do not play a role. In forthcoming work together with Marinelli, we show that the upper bound (1.1) can be transferred to a large class of stochastic convolutions and apply these new estimates to obtain improved well-posedness and regularity results for the associated stochastic evolution equations in L q -spaces.Let us roughly sketch our approach to (1.1). As a starting point, we use a classical result due to Meyer [37] and Yoeurp [53] to decompose the integrator as a sum of three localwhere M c is continuous, M q is purely discontinuous and quasi-left continuous and M a is purely discontinuous with accessible jumps. Sharp bounds for stochastic integrals with respect to continuous local martingales were already obtained in a more general setting [48].To estimate the integral with respect to M a , we prove, more generally, sharp bounds for an arbitrary purely discontinuous L q -valued local martingale with accessible jumps in Theorem 5.14. To establish this result, we first show that such a process can be represented as an essentially discrete object, namely a sum of jumps occurring at predictable times. Using an approximation argument, the problem can then be further reduced to proving Burkholder-Rosenthal type inequalities for L q -valued discrete-time martingales. In general, if 1 p < ∞ and X is a Banach space, we understand under Burkholder-Rosenthal inequalities estimates for X-valued martingale difference sequences (d i ) of the formStep 3: J is finite, ν is general. Without loss of generality we can assume that Eν(R + × J) < ∞. Then by a time-change argument as was used in the proof of Theorem B.3, we can assume that ν((s, t] × J) t − s a.s. for each t s 0, and apply Step 2.Step 4: J is general, ν is general. Without loss of generality assume that Eν(R + × J) < ∞. Let f be simple P-measurable, that is, there exist a K 1 and a partition J = J 1 ∪ · · · ∪ J K of J into disjoint sets such thatwhere f 1 , . . . , f k : R + × Ω → L q (S) are predictable. Fix j k ∈ J k , k = 1, . . . , K, and define J = {j 1 , . . . , j K }. Let ν be a new random measure on R + × Ω × J defined by ν(A × {j k }) = ν(A × J k ), A ∈ P, k = 1, . . . , K.