Grass roots vary in terms of depth and manner of spread depending on a variety of factors, including variety of grass, type of soil, and amount of available moisture. Generally, to be effective, borders should be cut and/or installed to a depth of 4-6 inches.  There are a variety of methods for installing and maintaining clean borders for flowerbeds in a lawn, including metal and plastic edgings, concrete or brick pavers, and simple cut edgings.  The type of edging you select will vary depending on the look you are trying to achieve with your garden spaces.  The least expensive method, and one often recommended by professional gardeners, is the simple cut-edge combined with liberal use of mulch, described very well in this Fine Gardening article.  Since installation of metal, plastic or other physical barrier borders also requires as a first step the clean, deep cutting of the turf surrounding the planting bed, once that is completed is a good time to assess whether that process alone provides the clean edge and appearance that you want for your own yard, or whether you prefer the possibly more finished look of one of the other edging materials.  Keep in mind that there are advantages and disadvantages to each of the edging methods.  The simple cut edge is least expensive and potentially easiest for mowing since the mower can ride right over the cut, therefore not requiring further trimming with other tools.  That edge will have to be refreshed once a year at least by recutting to keep the edge crisp and clean.  Plastic edgings usually have rolls at the top or other above-ground protuberances that will require trimming the abutting grass with a string trimmer or other mechanical means each time the lawn is mowed to keep it looking neat.  Concrete or brick pavers can be sunk into the ground so that the mower can ride over them, therefore not requiring separate trimming, but the spaces between the pavers will eventually host grasses and/or weeds which will have to be removed or killed to keep the neat appearance most people who use such borders want.  Keep in mind that with all these methods, some grasses will spread to your beds via seed and will still require removal by mechanical means throughout the growing season, but any of the edgings that you select, if properly maintained, should last a fair number of years and help you maintain a clean and neat definition between plant beds and lawn.