Over the past decade, there has been a growing interest in evolution equations of parabolic type that involve fractional-order derivatives in time of order in (0, 1). Such equations, also called subdiffusion equations, arise in various applications in engineering, physics, biology and finance. Hence, it is quite important to develop efficient and reliable computational tools for their numerical solution. For such problems, I will present a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds. Furthermore, pointwise-in-time a posteriori error bounds will be given, which lead to an adaptive time stepping algorithm with a local time step criterion.