Being atomically thin, flexible, and exhibiting considerable light emission and ultrafast non-equilibrium dynamics, semiconducting transition metal dichalcogenides (TMDs) have been considered as promising candidates for next-generation optoelectronic devices. The optical and electronic properties of TMDs are governed by a rich landscape of tightly bound excitons, including regular bright excitons, as well as optically inaccessible dark exciton states. Recently, strain engineering of monolayer TMDs has been introduced to tune their optical properties, such as the exciton transition energy, exciton-phonon coupling, or the Stokes shift [1-3].Transport of charge carriers is crucial for nanoelectronics. In conventional materials, electronic transport can be conveniently controlled by external electric fields. However, the tightly bound excitons, being neutral particles, are only weakly affected by electrical fields. We demonstrate that mechanical strain can also be used to manipulate the transport of excitons in TMDs. To this end, we apply homogeneous tensile strain to a WS2 monolayer by bending the substrate [1], which causes a redshift of the X0 exciton photoluminescence. By measuring the spatiotemporal photoluminescence after near-resonant excitation with femtosecond laser pulses, we map the spread of excitons and extract the strain-dependent diffusion coefficient [4].Furthermore, we demonstrate the propagation of excitons in an inhomogeneous strain landscape. We create inhomogeneous tensile strain in TMD monolayers by transferring them onto patterned substrates with nanopillars or by a nanoimprint technique [5]. Due to the redshift of the exciton resonances with applied strain, excitons are expected to move towards high-strain regions in an inhomogeneous strain field - the so-called "funneling" effect. We verify this behavior for the "bright" TMD material monolayer MoSe2. In the case of the "dark" monolayer WS2, we observe exactly the opposite effect. Here, the excitons are expelled from the high-strain regions ("anti-funneling") [6]. By comparing our experimental results with a microscopic theory, we explain this observation by the drift of momentum-dark KΛ excitons, which, in contrast to bright excitons, shift to higher energies with strain.Our joint experiment-theory study highlights the dominant role of momentum-dark excitons for the dynamics in monolayer TMDs and provides crucial design guidelines for TMD devices based on exciton transport.