To be precise, when we say that "X exists," we mean that the presently available, cumulative statistical database for experiments studying X, provides strong, scientifically credible evidence for repeatable, anomalous, X-like effects.
With this in mind, ESP exists, presentiment (physical changes in skin reactivity, pupil size, heart rate, and other factors indicating precognition before a stimulus is applied) exists, telepathy (direct mind-mind communication) exists, and mind-matter interaction (previously known as psychokinesis or PK) exists. The survival of bodily death remains unproven, though there is suggestive evidence for this from the reincarnation research performed by Ian Stevenson and others. (Note that we are using the terms ESP, telepathy and MMI in the technical sense, not in the popular sense. See What do parapsychologists study?)
ESP is statistically robust, meaning it can be reliably demonstrated through repeated trials. However, it may vary it but it tends to be weak when simple geometric symbols are used as targets. Photographic or video targets often produce effects many times larger, and there is some evidence that ESP on natural locations (as opposed to photos of them), and in natural contexts may be stronger still. Also, a lot has been learned about what kinds of conditions (such as the partial sensory deprivation used in the Ganzfeld) can enhance psi.
Some mind-matter interaction (MMI) effects have also been shown to exist. When individuals focus their intention on mechanical or electronic devices that fluctuate randomly, the fluctuations change in ways that conform to their mental intention. Under control conditions, when individuals direct their attention elsewhere, the fluctuations are in accordance with chance.
It should be noted that an increasing number of parapsychologists are moving beyond proof-oriented research (feeling that psi has already been sufficiently proven for anyone willing to actually read and consider the experimental research) to process-oriented, qualitative research. These studies are looking at a variety of factors (such as the kind of target used) to better understand these phenomena.