We study classification and regression for variable length sequential data, which is either non-uniformly sampled or contains missing samples. In most sequential data processing studies, one considers data sequence is uniformly sampled and complete, i.e., does not contain missing input values. However, non-uniformly sampled sequences and the missing data problem appear in a wide range of fields such as medical imaging and financial data. To resolve these problems, certain preprocessing techniques, statistical assumptions and imputation methods are usually employed. However, these approaches suffer since the statistical assumptions do not hold in general and the imputation of artificially generated and unrelated inputs deteriorate the model. To mitigate these problems, in chapter 2, we introduce a novel Long Short-Term Memory (LSTM) architecture. In particular, we extend the classical LSTM network with additional time gates, which incorporate the time information as a nonlinear scaling factor on the conventional gates. We also provide forward pass and backward pass update equations for the proposed LSTM architecture. We show that our approach is superior to the classical LSTM architecture, when there is correlation between time samples. In chapter 3, we investigate regression for variable length sequential data containing missing samples and introduce a novel tree architecture based on the Long Short-Term Memory (LSTM) networks. In our architecture, we employ a variable number of LSTM networks, which use only the existing inputs in the sequence, in a tree-like architecture without any statistical assumptions or imputations on the missing data. In particular, we incorporate the missingness information by selecting a subset of these LSTM networks based on presence-pattern of a certain number of previous inputs.