The thesis work discusses the design, realization, characterization and operation of a two-qubit processor implemented using capacitively coupled tunable superconducting qubits of the Transmon type. Each qubit can be manipulated and read out individually using a non-destructive single-shot readout. In addition, a universal-two qubit gate can be implemented using the interaction between the qubits. The processor implements therefore all basic building blocks of a universal two-qubit quantum processor. Using it, we implement the universal square root of iSWAP two-qubit gate, characterizing the gate operation by quantum process tomography and obtaining a gate fidelity of 90 %. We use this gate to create entangled two-qubit Bell states and perform a test of the CHSH Bell inequality, observing a violation of the classical boundary by 22 standard eviations after correcting for readout errors. Using the implemented two-qubit gate, we run the so-called Grover search algorithm: For two-qubits, this algorithm finds among four elements {00, 01, 10, 11} the one element y that solves a search problem encoded by a function f for which f(y) = 1 and f(x != y) = 0. Our implementation retrieves the correct answer to the search problem after a single evaluation of the search function f(x), with a success probability between 52 % and 67 %, therefore outperforming classical algorithms that are bound to a success probability of 25 %. This constitutes therefore a proof-of-concept of the quantum speed-up for superconducting quantum processors. Finally, we propose a scalable architecture for a superconducting quantum processor that can potentially overcome the scalability issues faced by today's superconducting qubit architectures.