Context. Lithium has long been recognized as a powerful tool for investigating the internal mixing of low-mass stars. In addition, accurate measurements of lithium abundances in young solar-type stars provide independent and reliable age diagnostics. Aims. To study the relationship between lithium and activity in normal solar type-type stars and determine the effectiveness of lithium and kinematics as activity indicators, we perform a lithium survey of nearby late-type stars. We search for nearby young solar-type stars and investigate the relationship between lithium abundances and X-ray activity by measuring log L x . Methods. On the basis of high-resolution spectroscopic observations, we derived the lithium abundances of 33 young solar-type stars by comparing the measured Li I λ 670.8 nm equivalent widths with the curve of growth calculations in non-LTE (NLTE) conditions. Results. We obtain the lithium abundances of a sample of young solar-type stars. Using the lithium abundance and X-ray luminosity, L x , we investigate the relationship between Li abundance and X-ray activity. We find a clear correlation between lithium abundance (log N(Li)) and X-ray luminosity, log L x , for our sample stars. Conclusions. As the X-ray luminosity, L x , becomes stronger, the lithium abundance decreases in our sample (11 pre-main-sequence, 15 zero-age-main-sequence, and 7 young main-sequence) stars. The results imply that there is a large number of solar-type stars. A correlation appears to exist between Li abundance and age, thus confirming the presence of very active young stars close to the Sun.