Implantable neuroelectronic interfaces have enabled breakthrough advances in the clinical diagnosis and treatment of neurological disorders, as well as in fundamental studies of brain function, behavior, and disease. Intracranial electroencephalography (EEG) mapping with stereo-EEG (sEEG) depth electrodes is routinely adopted for precise epilepsy diagnostics and surgical treatment, while deep brain stimulation has become the standard of care for managing movement disorders. Intracortical microelectrode arrays for high-fidelity recordings of neural spiking activity have led to impressive demonstrations of the power of brain-machine interfaces for motor and sensory functional recovery. Yet, despite the rapid pace of technology development, the issue of establishing a safe, long-term, stable, and functional interface between neuroelectronic devices and the host brain tissue still remains largely unresolved. A body of work spanning at least the last 15 years suggests that safe, chronic integration between invasive electrodes and the brain requires a close match between the mechanical properties of man-made components and the neural tissue. In other words, the next generation of invasive electrodes should be soft and compliant, without sacrificing biological and chemical stability. Soft neuroelectronic interfaces, however, pose a new and significant surgical challenge: bending and buckling during implantation that can preclude accurate and safe device placement. In this topical review, we describe the next generation of soft electrodes and the surgical implantation methods for safe and precise insertion into brain structures. We provide an overview of the most recent innovations in the field of insertion strategies for flexible neural electrodes such as dissolvable or biodegradable carriers, microactuators, biologically-inspired support structures, and electromagnetic drives. In our analysis, we also highlight approaches developed in different fields, such as robotic surgery, which could be potentially adapted and translated to the insertion of flexible neural probes.