Time-nonlocal evolution equations (EEs) are popular theoretical and applied models that introduce time-heterogeneity and memory effects in the classical heat equation. In this thesis we focus on wellposedness and Feynman-Kac formulae for several time-nonlocal EEs. In particular, we define and study a time-nonlocal initial condition, endowing it with probabilistic intuition. The main contribution is the stochastic-classical-wellposedness for the Marchaud EE and the stochastic-weak-wellposedness for the Marchaud-type EE. In the latter case, the kernel defining the nonlocal time derivative is allowed to be time-dependent. As a particular case we treat the Caputo-type inhomogeneous EE.