Evolutionary finance studies financial markets from an evolutionary point of view. A financial market can be interpreted in the context of its evolution: it can be understood as a dynamical system in which frequently interacting investment strategies compete for market capital. We are mainly interested in the long-run performance of investment strategies. This thesis explores the "Darwinian theory" of portfolio selection. An asset market can be modelled by a game-theoretic evolutionary model in which asset prices are endogenously determined by market clearing condition. A general version of the Kelly rule is shown to allow an investor to "survive" in the asset market. We then investigate the stochastic model with independent and identical distributed states of the world from a different, game-theoretic, angle and examine Nash equilibrium strategies, satisfying equilibrium conditions with probability one. Evolutionary finance and asset market games also provide new angles to present fundamental facts of capital growth theory. Relations between financial growth and the property of "survival" of investment strategies are established in the market selection process.