This book presents the latest research on global well-posedness including asymptotic behavior of solutions to some non-classical diffusion equations with fading memories, nonlocal terms or delays in several time-dependent spaces. The results collected in this book have been established by the authors and their collaborators over recent years. This book has two distinguishing features. First, while there are many published works on non-classical diffusion equations in Sobolev spaces without time-dependent terms but few results in time-dependent spaces, this book fills this gap. Second, this book provides new results on the existence, regularity and upper semicontinuity of time-dependent global attractors, strong attractors, and pullback attractors in time-dependent spaces, as well as the ideas and methods for dealing with these problems that can be used in other related models.