The objective of this paper is to study detectability, observability and related Lyapunov-type theorems of linear discrete-time time-varying stochastic systems with multiplicative noise. Some new concepts such as uniform detectability, K-infinity-exact detectability (resp. K-WFT-exact detectability, K-FT-exact detectability, K-N-exact detectability) and K-infinity-exact observability (resp. K-WFT-exact detectability, K-FT-exact detectability, K-N-exact detectability) are introduced, respectively, and nice properties associated with uniform detectability, exact detectability and exact observability are also obtained. Moreover, some Lyapunov-type theorems associated with generalised Lyapunov equations and exponential stability in mean square sense are presented under uniform detectability, K-N-exact observability and K-N-exact detectability, respectively.