I present a comprehensive investigation of fast stars in the Milky Way, from brisk disc stars to stars escaping the Galaxy. My thesis is that fast stars are the smoking guns of extreme stellar collisions and explosions, and so can act as an intermediary to studying these theoretically-unconquered astrophysical processes.

In Chapter 1 I give a history of fast stars, address what it means for a star to be fast, and describe the processes that accelerate stars. I concisely summarise the Gaia mission, whose recent data releases heavily influenced this thesis.

Supernovae in binary systems can fling away the companion; if a runaway companion can be associated with a supernova remnant, then together they reveal the evolution that led to the supernova. However, these associations are difficult to establish. In Ch. 2, I develop a sophisticated Bayesian methodology to search the nearest ten remnants for a companion, by combining data from Gaia DR1 with a 3D dust-map and binary population synthesis. With Gaia DR2, I will identify companions of tens of supernova remnants and thus open a new window to studying late-stage stellar evolution.

It is unknown why 17% of B stars are spinning near break-up; these stars are termed Be stars because of emission lines from their ejected material. Their rapid spin could be due to mass transfer, but in Ch. 3 I show this would create runaway Be stars. I demonstrate using a hierarchical Bayesian model that these exist in sufficient numbers, and thus that all Be stars may arise from mass transfer.

The stars escaping the Milky Way are termed hypervelocity stars. In Ch. 4, I overturn the consensus that the hypervelocity stars originated in the Galactic centre by showing that a Large Magellanic Cloud (LMC) origin better explains their distribution on the sky. In Ch. 5 I present three ground-breaking hypervelocity results with Gaia DR2: 1) only 41 of the 524 hypervelocity star candidates are truly escaping, 2) at least one of the hypervelocity stars originates in the LMC, and 3) the discovery of three hypervelocity white dwarf runaways from thermonuclear supernovae.