This paper develops a general theory on rates of convergence of penalized spline estimators for function estimation when the likelihood functional is concave in candidate functions, where the likelihood is interpreted in a broad sense that includes conditional likelihood, quasi-likelihood, and pseudo-likelihood. The theory allows all feasible combinations of the spline degree, the penalty order, and the smoothness of the unknown functions. According to this theory, the asymptotic behaviors of the penalized spline estimators depends on interplay between the spline knot number and the penalty parameter. The general theory is applied to obtain results in a variety of contexts, including regression, generalized regression such as logistic regression and Poisson regression, density estimation, conditional hazard function estimation for censored data, quantile regression, diffusion function estimation for a diffusion type process, and estimation of spectral density function of a stationary time series. For multi-dimensional function estimation, the theory (presented in the Supplementary Material) covers both penalized tensor product splines and penalized bivariate splines on triangulations.