Metropolis-Hastings algorithms are a common type of Markov Chain Monte Carlo method for sampling from a desired probability distribution. In this talk, I will present a general framework for such algorithms which is based on a fundamental involution structure on a general state space, and encompasses several popular algorithms as special cases, both in the finite- and infinite-dimensional settings. In particular, these include random walk, preconditioned Crank-Nicolson (pCN), schemes based on a suitable Langevin dynamics such as the Metropolis Adjusted Langevin algorithm (MALA), and also ones based on Hamiltonian dynamics including several variants of the Hamiltonian Monte Carlo (HMC) algorithm. In fact, with a slight generalization of our first framework, we are also able to cover algorithms that generate multiple proposals at each iteration. These have the potential of providing efficient sampling schemes through the use of modern parallel computing resources. Here we derive several generalizations of the aforementioned algorithms following as special cases of this multiproposal framework. To illustrate the effectiveness of these sampling procedures, we present applications in the context of some Bayesian inverse problems in fluid dynamics. This is based on joint works with N. Glatt-Holtz (Indiana University), A. Holbrook (UCLA), and J. Krometis (Virginia Tech).