The aim of this thesis is to identify practical applications where quantum computing could have an advantage over what is achievable with conventional means of computing, and what advances are needed in order to actualize that potential. We investigate the possibilities on both quantum annealing devices (analogue quantum computers) as well as on gate-based devices (digital quantum computers). Quantum annealing devices in particular have grown in size and capabilities over the last decade. While they are natural candidates for providing improved solvers for NP-complete optimization problems, the number of qubits, accuracies and available couplings are still limited and their aptitude has yet to be confirmed. We use Monte Carlo methods and conceive an adaptive annealing schedule to assess the options to leverage them for the construction of satisfiability filters. We furthermore investigate their prospective as heuristic solvers for the traveling salesman problem, and what future enhancements in terms of geometries and couplings would be beneficial for that. Based on our simulations there is no reason to expect any benefits to leveraging analogue quantum computers over state-of-the-art classical methods. However, we see an implementation of annealing schemes on future digital devices as a promising approach that doesn’t suffer from the same issues that impede performance on analogue devises. To that effect, we construct and implement an efficient quantization of the Metropolis-Hastings algorithm. Opposed to following the common way of quantization à la Szegedy that is usually defined with respect to an oracle, we reformulate the walk to closely mimic the classical algorithm and thus circumvent having to rely on costly quantum arithmetics. Our proposed realization thereby can lead to substantial savings. While theoretical arguments promise a quadratic speedup in the asymptotic limit, we numerically confirm that a polynomial speedup in terms of minimal total time to solution can be achieved for pragmatic use. We explore the prospects of using quantum walks in a heuristic setting and estimate the gate times the would be required to outperform a classical supercomputer. Finally, we elaborate on the role of programming languages, and how software tools can accelerate the advancement of the field. We discuss unique aspects of quantum programming and the purpose of Q# in particular, and conclude by highlighting what developments are needed for quantum computing to live up to its potential.