  This paper is devoted to the investigation of gradient flows in asymmetric
metric spaces (for example, irreversible Finsler manifolds and Minkowski normed
spaces) by means of discrete approximation. We study basic properties of curves
and upper gradients in asymmetric metric spaces, and establish the existence of
a curve of maximal slope, which is regarded as a gradient curve in the
non-smooth setting. { Introducing} a natural convexity assumption on the
potential function, { which is called the $(p,\lambda)$-convexity,} we also
obtain some regularizing effects on the asymptotic behavior of curves of
maximal slope. Applications include several existence results for gradient
flows in Finsler manifolds, doubly nonlinear differential evolution equations
on { infinite-dimensional Funk spaces}, and heat flow on compact Finsler
manifolds.
