A method based on the physics-informed neural networks (PINN) is proposed to solve the transient nonlinear heat conduction problems and estimate the temperature-dependent thermal conductivity. Firstly, a loss function is formulated with the residual error of partial differential equation, initial conditions and boundary conditions for the heat conduction problems. Then, the automatic differentiation is applied to acquire the partial derivatives of temperature in the partial differential equation. The heat conduction problems are solved by applying gradient descent algorithm to update the network parameters and to minimize the loss function. The influences of different numbers of hidden layers, neurons and interior collection points on the results are discussed. Finally, the PINN is applied to identify the temperature-dependent thermal conductivities. The loss function is formulated with the residual error of governing equation, measured temperature and computed temperature. The network parameters and thermal conductivity are updated by the gradient descent algorithm to approximate the exact solution. The influences of different measurement points and errors on results are also compared. The results show that the proposed method is an effective approach to solve the transient heat conduction problem and estimate the temperature-dependent thermal conductivity.