Many processes in physics and biology can be understand through the framework of escape from a metastable state, including (but not limited to) the rates of chemical reactions, the unfolding of proteins, the nucleation of bubbles, and the condensation of gases. To understand the kinetics of these processes, we have to be able to calculate the rate of escape. In this thesis, I solve several of such of escape problems, each addressing a specific physical or biological system. I first show how the forced unfolding of heteropolymers could be a process with non-exponential kinetics, developing ideas about the importance of unfolding pathways in determining kinetics of unfolding. Then, I consider forced unfolding when a molecule is attached to a yielding (viscoelastic) substrate, and a constant force is applied. I show that the rates of unfolding depend on both the elastic and viscous response of the substrate. This problem is related to the biological process of mechanosensing, when the unfolding `sensor' protein exposes catalytic residues and generates a chemical signal to the cell. Related to this is the analysis of population-dynamics study of cells adhesion on substrates, which allows me to extract key characteristics and parameters of the adhesome complex. Then, I apply the ideas of escape from a metastable state to ask about the rates of a ligand at the end of a tethered polymer binding to a surface receptor, using a mean field approach to reduce the problem to one dimension. I show that there is a trade-off between the entropic cost of reaching to a receptor vs the volumetric cost of expanding the tether length. I then show that for a Gaussian chain with multiple ligands along its length, there exists a finite, non-zero optimal number of ligands to minimise the time taken for the end of the chain to bind to the surface. Finally, I consider the problem of microswimmers in an obstacle lattice, calculating their transport properties, and showing how we can use lattices to examine the underlying stochastic dynamics.