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Abstract: Covariance is a fundamental measure to describe the dependency structure between two random variables and covariance modelling is significant in a wide range of applications. Existing systems mainly assume the covariance is static even though covariance structure is typically dynamic in real-world settings, which indicates the need of dynamic covariance modelling. Two main approaches are mainly adopted in covariance parameterisation: covariance-function-based methods and matrix-decomposition-based methods. Covariance-function-based approaches describe the covariance structure with parsimonious parameters, thereby effectively solve the the rapid growth in the number of parameters with the increment of covariance dimension but they may fail to guarantee positive definiteness in non-stationary settings. Covariance decomposition methods, on the other hand, offer unconstrained factors that guarantee positive-definite output but suffer from the quadratic increase of estimated parameters and poor statistical interpretability. As a result, methods that simultaneously ensure positive definiteness and maintain parsimonious parameterisations remain largely unexplored. Motivated by these limitations, we propose a flexible estimation framework for covariance functions by adopting positive-definite regularisations of the objective functions. This framework imposes penalties on eigenvalues or the determinant of the estimated covariance matrix, thereby enforcing positive-definiteness with no additional estimated parameters required. This framework effectively addresses the positive-definite issue for covariance functions while maintaining its strengths, which makes it suitable for high-dimensional dynamic covariance modelling. Simulation studies and real-data applications demonstrate that the proposed framework consistently improves estimation accuracy while ensuring positive definite covariance estimates across a range of stationary and non-stationary covariance structures.