Scoring rules are an established way to compare predictive performance between model classes. In the context of survival analysis, they require adaptation in order to accommodate censoring and other aspects specific to survival tasks. This work investigates the use of scoring rules for model training rather than evaluation. Doing so, we establish a general framework for training survival models that is model-agnostic and can learn event time distributions parametrically or non-parametrically. In addition, our framework is not restricted to any specific scoring rule. Although we focus on neural network-based implementations, we also provide proof-of-concept implementations using gradient boosting, generalized additive models, and trees. Empirical comparisons on synthetic and real-world data indicate that scoring rules can be successfully incorporated into model training and yield competitive predictive performance with established time-to-event models.