While Compustat and CRSP both have CUSIP and Ticker symbol variables, there is no common, unique identifier for linking or merging CRSP & COMPUSTAT data.
CRSP does sell a CCM product that has a linking table that gives every 'linkable' GVKEY (Compustat ID key) links to CRSP PERMCO and PERMNO Id keys. Often this linking or mapping is not unique, as over different date ranges the appropriate PERMCO/PERMNO changes.
Traditionally, researchers that do not have CCM access build a linking algorithm that compares CRSP CUSIP to Compustat CUSIP, but this method has various problems and is not 100% accurate. We built refinements to combine any two sets of CUSIP and/or Ticker Symbol.