When people talk about perspective usually they are thinking of linear perspective. But there are many more ways to show depth in a drawing or painting. There is also atmospheric perspective, color perspective, and size perspective. Linear perspective is used most commonly when drawing or painting architecture or something with a lot of lines. This system was carefully worked out by the renaissance artists, and we still use it today. 

Artists in the far east show depth in their art by more subtle means. Notice that you will see some artwork that shows depth simply by the size and placement of the elements. Nothing more is needed. The nearer the thing, the larger, and vice versa. Likewise showing more or less detail can sometimes be enough to show depth in a scene. 

A clear understanding of color theory is also helpful in showing depth in your paintings. Cool colors recede and warm colors come forward. So one way to show depth is to use cool colors in the background. Cool colors are primarily blues and greens. And then show warm colors in the foreground (reds, oranges, yellows, most browns).

 Greys are your true neutrals and can be either warm or cool. Familiarize yourself with the color wheel and create your own color chart for a deeper understanding of color.