  The aim of the paper is to provide a series of new examples of smooth
surfaces in P^4, not of general type, in degrees varying from 12 up to 14, and
to describe their geometry. By using mainly syzygies and liaison techniques, we
construct the following families of surfaces:
  - minimal proper elliptic surfaces of degree 12 and sectional genus 13,
  - two types of non-minimal proper elliptic surfaces of degree 12 and
sectional genus 14,
  - non-minimal K3 surfaces of degree 13 and sectional genus 16, and -
non-minimal K3 surfaces of degree 14 and sectional genus 19.
