In this thesis, we consider three different market impact models and their regularity. Regularity of a market impact model is characterized by properties of optimal liquidation strategies. Specifically, we discuss absence of price manipulation and absence of transaction-triggered price manipulation. Moreover, we introduce a new regularity condition called positive expected liquidation costs. The first market impact model under consideration allows for transient impact with a time-dependent liquidity parameter. This includes time-dependent permanent impact as a special case. In this model, we show an example for an arbitrage opportunity while the unaffected price process is a martingale. Furthermore, we show that regularity may depend strongly on the liquidation time horizon. We also find that deterministic strategies can be suboptimal for a risk-neutral investor even if the liquidity parameter is a martingale. Second, we extend an Almgren-Chriss model with the possibility to trade in a dark pool. In particular, we model the cross impact of trading in the exchange onto prices in the dark pool and vice versa. We find that the model is regular if there is no temporary cross impact from the exchange to the dark pool, full permanent cross impact from the dark pool to the exchange, and an additional penalization of orders executed in the dark pool. In other cases, we show by a number of examples how the regularity depends on the interplay of all model parameters and on the liquidation time constraint. Third, we consider a linear transient impact model in discrete time with the possibility to trade multiple assets including cross impact between the different assets. The model is regular if the matrix-valued decay kernel of market impact is a positive definite function. We characterize both symmetric and non-symmetric matrix-valued positive definite functions. We discuss nonnegative and nonincreasing decay kernels. If a decay kernel is additionally symmetric and convex, it is positive definite. Moreover, if it is also commuting, we show that the optimal discrete-time strategies converge to an optimal continuous-time strategy. For matrix-valued exponential functions, we provide explicit solutions.