On debian based distributions,apt is the go-to way to perform all packages operations. dpkg is as well very useful, especially when you build your own .deb packages.

Advanced Packaging Tool

# list installed packages
apt-cache pkgnames
# including missing deps
apt-cache --all-names pkgnames

# show information on a package
apt-cache showpkg $pkg

# search available packages to install
apt-cache search $regexp
# with full info
apt-cache search --full $regexp

# show a package record
apt-cache show $pkg

# show package sources
apt-cache showsrc $pkg

# show dependencies
apt-cache depends $pkg

# --recurse to make them recursive

# show reverse dependencies
# to filter with the following options
# --no-pre-depends
# --no-depends
# --no-recommends
# --no-suggests
# --no-conflicts
# --no-breaks
# --no-replaces
# --no-enhances
apt-cache rdepends $pkg

# show policy
apt-cache policy $pkg

# cache stats
apt-cache stats

# show every package cached
apt-cache dump
# more info
apt-cache dumpavail

# show unmet dependencies
apt-cache unmet
# filter with --important