One could probably find a bunch through a Google search but I couldn't name a startup in the law space off the top of my head. I also don't remember ever seeing one making headlines on tech blogs.
With all the enormous fees and the insane amount of paperwork I'd think it's a space that's screaming for new ideas but from an outside perspective it doesn't seem like there has been much innovation since MS Word... Has anyone got some insight?
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What, exactly, is a law start-up? There is a lot of new tech available to lawyers to make their jobs easier and to increase their productivity. The American Bar Association has committees that review new products every year.
Perhaps you don't sense that there are many legal "startups" because they don't tend to co-exist with the rest of the start-up community. Once they have scaled, you may not find them advertising or soliciting for you to be a client due to ethical rules that limit how and when attorneys can do this. Also, these startups tend to co-exit with other law firms in the legal sector rather than in the startup community. To give you a couple of examples:
Legal startup services that exist and grow online. Some people consider these innovative because you don't deal with an hourly rate (change in billing format), but one issue can be receiving cookie cutter services with less of catering to your specific needs. I think many of these startups have already scaled to national level because they're easily found on google.
I hope this gives you a little bit insight into the mix of startups and the law. Please let me know if you were thinking of a specific concept that differs from the examples I provided above.
Google "LegalTech" and eventually add "landscape" and you will find tons of startups in the space. Goes from NDA and contract automation to A.I. driven analysis of legal decisions made to extract how the practicioners apply the law to inferring the structure of a legal case from the facts provided (subsumption technology) to intelligent search for legal cases relating to a particualr issue, to analyzing trends among judges in different regions and geographies.