The Boulby Dark Matter Collaboration is running a WIMP Dark Matter research programme in the underground laboratory of Boulby Mine for over a decade. Results have been reported previously from the NaIAD detector array -using scintillation properties in NaI crystals [4,5,6]. Currently, the programme is based on (1) liquid Xenon (LXe) as the WIMP target and (2) directional detection in low pressure gas detectors (Drift, reported at this conference [3]). This contribution concentrates on Zeplin-1, the LXe first project: a 3.1kg of LXe scintillation detector with a background discrimination based on Pulse Shape Analysis. The current status of the experiment is shown and first limits on nuclear recoil events are reported. Setups with improved background discrimination tools are commissioned: Zeplin-2 and Zeplin-3 will be installed underground early 2004. Progress are reported in another contribution to this conference [1]. These are important steps towards the design of a ton-scale LXe Dark Matter detector array (reported in [2]).