Reservoir simulation is widely applied for the management of oil and gas production and CO2 storage operations. It can be computationally expensive, however, particularly when the flow physics is complicated and many simulation runs must be performed. This has motivated the development of reduced-order modeling (ROM) procedures, where the goal is to achieve high degrees of computational speedup along with reasonable solution accuracy. In this work we develop and apply two types of ROM methods -- one based on proper orthogonal decomposition (POD) and piecewise linearization, and one based on deep learning. We first develop a POD-based ROM, referred to as POD-TPWL, to simulate coupled flow-geomechanics problems. In POD-TPWL, proper orthogonal decomposition, which enables the representation of solution unknowns in a low-dimensional subspace, is combined with trajectory piecewise linearization (TPWL), where solutions with new sets of well controls are represented via linearization around previously simulated (training) solutions. The over-determined system of equations is projected into the low-dimensional subspace using a least-squares Petrov-Galerkin procedure. The states and derivative matrices required by POD-TPWL, generated by an extended version of Stanford's Automatic-Differentiation-based General Purpose Research Simulator, are provided in an offline (pre-processing or training) step. Offline computational requirements correspond to the equivalent of 5-8 full-order simulations, depending on the number of training runs used. Runtime (online) speedups of O(100) or more are achieved for new POD-TPWL test-case simulations. The POD-TPWL model is tested extensively for a 2D coupled problem involving oil-water flow and geomechanics. It is shown that POD-TPWL provides predictions of reasonable accuracy, relative to full-order simulations, for well-rate quantities, global pressure and saturation fields, global maximum and minimum principal stress fields, and the Mohr-Coulomb rock failure criterion, for the cases considered. A systematic study of POD-TPWL error is conducted using various training procedures for different levels of perturbation between test and training cases. The use of randomness in the well bottom-hole pressure profiles used in training is shown to be beneficial in terms of POD-TPWL solution accuracy. The procedure is also successfully applied to a prototype 3D example case. We next apply the POD-TPWL reduced-order modeling framework to simulate and optimize the injection stage of CO2 storage operations. The use of multiple derivatives, meaning that the linearizations are performed around different training solutions at different time steps, is described and assessed. Two example cases are presented, and the ability of the POD-TPWL model to accurately capture bottom-hole pressure, when time-varying CO2 injection rates are prescribed, is demonstrated. It is also shown that, for these examples, the reduced-order models can provide accurate estimates of CO2 molar fraction at particular locations in the domain. The POD-TPWL model is then incorporated into a mesh adaptive direct search optimization framework where the objective is to minimize the amount of CO2 reaching a target layer at the end of the injection period. The POD-TPWL model is shown to be well suited for this purpose and to provide optimization results that are comparable to those obtained using full-order simulations. POD-TPWL preprocessing computations entail a (serial) time equivalent of about 6.7 full-order simulations, though the resulting runtime speedups, relative to full-order simulation, are about 100--150 for the cases considered. Finally, we develop a new deep-learning-based ROM for reservoir simulation. The reduced-order model is based on an existing embed-to-control (E2C) framework and includes an auto-encoder, which projects the system to a low-dimensional subspace, and a linear transition model, which approximates the evolution of the system states in low dimension. In addition to the loss function for data mismatch considered in the original E2C framework, we introduce a physics-based loss function that penalizes predictions that are inconsistent with the governing flow equations. The loss function is also modified to emphasize accuracy in key well quantities of interest (e.g., fluid production rates). The E2C ROM is shown to have interesting parallels with POD-TPWL. The new ROM is applied to oil-water flow in a 2D heterogeneous reservoir. A total of 300 high-fidelity training simulations are performed in the offline stage, and the network training requires 10-12~minutes on a Tesla V100 GPU node. Online (runtime) computations achieve speedups of O(1000) relative to full-order simulations. Extensive test case results, with well controls varied over large ranges, are presented. Accurate ROM predictions are achieved for global saturation and pressure fields at particular times, and for injection and production well responses as a function of time. Error is shown to increase when 100 or 200 (rather than 300) training runs are used to construct the E2C ROM.